TPTP Problem File: ITP276^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP276^2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Uniqueness 00216_013627
% 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_00216_013627 [Des22]
% Status : ContradictoryAxioms
% Rating : 0.67 v8.1.0
% Syntax : Number of formulae : 9271 (2803 unt; 863 typ; 0 def)
% Number of atoms : 26872 (9292 equ; 1 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 180928 (2245 ~; 341 |;2130 &;163810 @)
% ( 0 <=>;12402 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 8 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 5388 (5388 >; 0 *; 0 +; 0 <<)
% Number of symbols : 853 ( 849 usr; 25 con; 0-8 aty)
% Number of variables : 29652 (2946 ^;25067 !; 784 ?;29652 :)
% ( 855 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_CAX_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 15:41:56.283
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
thf(ty_t_Record_Otuple__isomorphism,type,
tuple_isomorphism: $tType > $tType > $tType > $tType ).
thf(ty_t_VEBT__Definitions_OVEBT,type,
vEBT_VEBT: $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Old__Datatype_Onode,type,
old_node: $tType > $tType > $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Typerep_Otyperep,type,
typerep: $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 (840)
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_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_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom,type,
semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__divide,type,
idom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__modulo,type,
idom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Transcendental_Oln,type,
ln:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : $o ).
thf(sy_cl_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_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_Onormalization__semidom,type,
normal8620421768224518004emidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot2__space,type,
topological_t2_space:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_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_Real__Vector__Spaces_Ometric__space,type,
real_V7819770556892013058_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__algebra,type,
real_V6157519004096292374lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo8386298272705272623_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo7287701948861334536_space:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__semigroup__mult,type,
topolo4211221413907600880p_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
real_V8037385150606011577_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__algebra__1,type,
real_V2191834092415804123ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
semiri2026040879449505780visors:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Oorder__topology,type,
topolo2564578578187576103pology:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
thf(sy_cl_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_Olinorder__topology,type,
topolo1944317154257567458pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo4958980785337419405_space:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V822414075346904944vector:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oordered__real__vector,type,
real_V5355595471888546746vector:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra,type,
real_V4412858255891104859lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1,type,
real_V2822296259951069270ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra,type,
real_V8999393235501362500lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Onormalization__semidom__multiplicative,type,
normal6328177297339901930cative:
!>[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__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_Ocinfinite,type,
bNF_Ca4139267488887388095finite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
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_OnatLess,type,
bNF_Ca8459412986667044542atLess: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
bNF_Ca7133664381575040944arCard:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Def_OGrp,type,
bNF_Grp:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
thf(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > 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__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__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__Wellorder__Constructions_OFunc,type,
bNF_Wellorder_Func:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( A > B ) ) ) ).
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_Ocurr,type,
bNF_Wellorder_curr:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ A ) > ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
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__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__BNF__LFPs_Osum_Osize__sum,type,
basic_BNF_size_sum:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( sum_sum @ 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_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > B > $o ) > ( C > D > $o ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) > $o ) ).
thf(sy_c_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_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Code__Numeral_OSuc,type,
code_Suc: code_natural > code_natural ).
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_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
code_nat_of_natural: code_natural > nat ).
thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
code_natural_of_nat: nat > code_natural ).
thf(sy_c_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_Ofixp,type,
comple115746919287870866o_fixp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates,type,
comple6359979572994053840erates:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
comple7512665784863727008ratesp:
!>[A: $tType] : ( ( A > A ) > A > $o ) ).
thf(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Omonotone,type,
comple7038119648293358887notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $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_Conditionally__Complete__Lattices_Opreordering__bdd,type,
condit622319405099724424ng_bdd:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd,type,
condit16957441358409770ng_bdd:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Countable_Ofrom__nat,type,
from_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Countable_Onat__to__rat__surj,type,
nat_to_rat_surj: nat > rat ).
thf(sy_c_Countable_Onth__item,type,
nth_item:
!>[A: $tType] : ( nat > ( set @ ( old_node @ A @ product_unit ) ) ) ).
thf(sy_c_Countable_Onth__item__rel,type,
nth_item_rel: nat > nat > $o ).
thf(sy_c_Countable__Set_Ocountable,type,
countable_countable:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Deriv_Odifferentiable,type,
differentiable:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__derivative,type,
has_derivative:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__field__derivative,type,
has_field_derivative:
!>[A: $tType] : ( ( A > A ) > A > ( filter @ A ) > $o ) ).
thf(sy_c_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_Oequivp,type,
equiv_equivp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
thf(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Extended__Nat_OeSuc,type,
extended_eSuc: extended_enat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_OAbs__enat,type,
extended_Abs_enat: ( option @ nat ) > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
extended_case_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
extend4730790105801354508finity:
!>[A: $tType] : A ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofiltercomap,type,
filtercomap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofiltermap,type,
filtermap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ).
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_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofolding__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_Fun_Ostrict__mono__on,type,
strict_mono_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_OGcd__class_OLcm,type,
gcd_Lcm:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > ( product_prod @ int @ int ) ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Obounded__quasi__semilattice,type,
bounde8507323023520639062attice:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
thf(sy_c_GCD_Obounded__quasi__semilattice__set,type,
bounde6485984586167503788ce_set:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
bounde2362111253966948842tice_F:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( set @ A ) > A ) ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__class_Olcm,type,
gcd_lcm:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
semiri4206861660011772517g_char:
!>[A: $tType] : ( ( itself @ A ) > nat ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
semiring_gcd_Lcm_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ogroup,type,
group:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
thf(sy_c_Groups_Ogroup__axioms,type,
group_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Omonoid__axioms,type,
monoid_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
groups1027152243600224163dd_sum:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
groups1962203154675924110t_prod:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set,type,
groups778175481326437816id_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
groups_comm_monoid_F:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
groups_comm_monoid_G:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list,type,
groups1828464146339083142d_list:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list__set,type,
groups4802862169904069756st_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__list,type,
groups_monoid_list:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Omonoid__list_OF,type,
groups_monoid_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
groups5270119922927024881d_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_Hilbert__Choice_Obijection,type,
hilbert_bijection:
!>[A: $tType] : ( ( A > A ) > $o ) ).
thf(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
complete_lattice_gfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: ( product_prod @ nat @ nat ) > int ).
thf(sy_c_Int_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,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__order,type,
semilattice_order:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder_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__max,type,
lattices_ord_arg_max:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
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__max,type,
lattic501386751176901750rg_max:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
lattic5214292709420241887eutr_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
lattic4895041142388067077er_set:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set,type,
lattic149705377957585745ce_set:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF,type,
lattic1715443433743089157tice_F:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Limits_OBfun,type,
bfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Limits_OZfun,type,
zfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Limits_Oat__infinity,type,
at_infinity:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_Oarg__min__list__rel,type,
arg_min_list_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( list @ A ) ) > ( product_prod @ ( A > B ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
thf(sy_c_List_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_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olinorder_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__all,type,
list_all:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
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_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_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > int > ( list @ int ) ).
thf(sy_c_List_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__le,type,
map_le:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > $o ) ).
thf(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__upds,type,
map_upds:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_Ofunpow,type,
funpow:
!>[A: $tType] : ( nat > ( A > A ) > A > A ) ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
thf(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
rec_set_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T > $o ) ).
thf(sy_c_Nat_Osemiring__1__class_ONats,type,
semiring_1_Nats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Oint__decode,type,
nat_int_decode: nat > int ).
thf(sy_c_Nat__Bijection_Oint__encode,type,
nat_int_encode: int > nat ).
thf(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: nat > ( list @ nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: nat > nat > $o ).
thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: ( list @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Osum__decode,type,
nat_sum_decode: nat > ( sum_sum @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Osum__encode,type,
nat_sum_encode: ( sum_sum @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Onat__of__num,type,
nat_of_num: num > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Ois__num,type,
neg_numeral_is_num:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
thf(sy_c_Num_Onum_Orec__num,type,
rec_num:
!>[A: $tType] : ( A > ( num > A > A ) > ( num > A > A ) > num > A ) ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Osqr,type,
sqr: num > num ).
thf(sy_c_Old__Datatype_OAtom,type,
old_Atom:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ nat ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OIn0,type,
old_In0:
!>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OIn1,type,
old_In1:
!>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OLeaf,type,
old_Leaf:
!>[A: $tType,B: $tType] : ( A > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_ONode,type,
old_Node:
!>[B: $tType,A: $tType] : ( set @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
thf(sy_c_Old__Datatype_ONumb,type,
old_Numb:
!>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OPush,type,
old_Push:
!>[B: $tType] : ( ( sum_sum @ B @ nat ) > ( nat > ( sum_sum @ B @ nat ) ) > nat > ( sum_sum @ B @ nat ) ) ).
thf(sy_c_Old__Datatype_OPush__Node,type,
old_Push_Node:
!>[B: $tType,A: $tType] : ( ( sum_sum @ B @ nat ) > ( old_node @ A @ B ) > ( old_node @ A @ B ) ) ).
thf(sy_c_Old__Datatype_OScons,type,
old_Scons:
!>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_Ondepth,type,
old_ndepth:
!>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > nat ) ).
thf(sy_c_Old__Datatype_Onode_OAbs__Node,type,
old_Abs_Node:
!>[B: $tType,A: $tType] : ( ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) > ( old_node @ A @ B ) ) ).
thf(sy_c_Old__Datatype_Onode_ORep__Node,type,
old_Rep_Node:
!>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
thf(sy_c_Old__Datatype_Ontrunc,type,
old_ntrunc:
!>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Option_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( option @ A ) > ( A > ( option @ B ) ) > ( option @ B ) ) ).
thf(sy_c_Option_Ocombine__options,type,
combine_options:
!>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
thf(sy_c_Option_Ois__none,type,
is_none:
!>[A: $tType] : ( ( option @ A ) > $o ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ).
thf(sy_c_Option_Ooption_Opred__option,type,
pred_option:
!>[A: $tType] : ( ( A > $o ) > ( option @ A ) > $o ) ).
thf(sy_c_Option_Ooption_Orec__option,type,
rec_option:
!>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ).
thf(sy_c_Option_Ooption_Orel__option,type,
rel_option:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( option @ A ) > ( option @ B ) > $o ) ).
thf(sy_c_Option_Ooption_Oset__option,type,
set_option:
!>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
thf(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Order__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_Oabove,type,
order_above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > 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,type,
ordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Oordering__axioms,type,
ordering_axioms:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Oordering__top__axioms,type,
ordering_top_axioms:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Opartial__preordering,type,
partial_preordering:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Opreordering,type,
preordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Opreordering__axioms,type,
preordering_axioms:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Power_Opower_Opower,type,
power2:
!>[A: $tType] : ( A > ( A > A > A ) > A > nat > A ) ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc5280177257484947105e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > 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_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
product_rec_set_bool:
!>[T: $tType] : ( T > T > $o > T > $o ) ).
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_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( T > product_unit > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
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_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Random_Oiterate,type,
iterate:
!>[B: $tType,A: $tType] : ( code_natural > ( B > A > ( product_prod @ B @ A ) ) > B > A > ( product_prod @ B @ A ) ) ).
thf(sy_c_Random_Oiterate__rel,type,
iterate_rel:
!>[B: $tType,A: $tType] : ( ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > $o ) ).
thf(sy_c_Random_Olog,type,
log: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Ominus__shift,type,
minus_shift: code_natural > code_natural > code_natural > code_natural ).
thf(sy_c_Random_Onext,type,
next: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random_Opick,type,
pick:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > code_natural > A ) ).
thf(sy_c_Random_Orange,type,
range: code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random_Oselect,type,
select:
!>[A: $tType] : ( ( list @ A ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Rat_OAbs__Rat,type,
abs_Rat: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
field_char_0_Rats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Opcr__rat,type,
pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
thf(sy_c_Rat_Opositive,type,
positive: rat > $o ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oratrel,type,
ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Real_ORatreal,type,
ratreal: rat > real ).
thf(sy_c_Real_OReal,type,
real2: ( nat > rat ) > real ).
thf(sy_c_Real_Ocauchy,type,
cauchy: ( nat > rat ) > $o ).
thf(sy_c_Real_Opcr__real,type,
pcr_real: ( nat > rat ) > real > $o ).
thf(sy_c_Real_Opositive,type,
positive2: real > $o ).
thf(sy_c_Real_Orealrel,type,
realrel: ( nat > rat ) > ( nat > rat ) > $o ).
thf(sy_c_Real_Orep__real,type,
rep_real: real > nat > rat ).
thf(sy_c_Real_Ovanishes,type,
vanishes: ( nat > rat ) > $o ).
thf(sy_c_Real__Vector__Spaces_OReals,type,
real_Vector_Reals:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__bilinear,type,
real_V2442710119149674383linear:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear,type,
real_V3181309239436604168linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear__axioms,type,
real_V4916620083959148203axioms:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Oconstruct,type,
real_V4425403222259421789struct:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B ) ).
thf(sy_c_Real__Vector__Spaces_Odependent,type,
real_V358717886546972837endent:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Odim,type,
real_Vector_dim:
!>[A: $tType] : ( ( set @ A ) > nat ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V557655796197034286t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oextend__basis,type,
real_V4986007116245087402_basis:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Real__Vector__Spaces_Olinear,type,
real_Vector_linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
real_V7770717601297561774m_norm:
!>[A: $tType] : ( A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oof__real,type,
real_Vector_of_real:
!>[A: $tType] : ( real > A ) ).
thf(sy_c_Real__Vector__Spaces_Orepresentation,type,
real_V7696804695334737415tation:
!>[A: $tType] : ( ( set @ A ) > A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V8093663219630862766scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Real__Vector__Spaces_Ospan,type,
real_Vector_span:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Real__Vector__Spaces_Osubspace,type,
real_Vector_subspace:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Record_Otuple__isomorphism_OTuple__Isomorphism,type,
tuple_1188178415141063261rphism:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( ( product_prod @ B @ C ) > A ) > ( tuple_isomorphism @ A @ B @ C ) ) ).
thf(sy_c_Record_Otuple__isomorphism_Osize__tuple__isomorphism,type,
tuple_9181185373184732606rphism:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > nat ) > ( B > nat ) > ( C > nat ) > ( tuple_isomorphism @ A @ B @ C ) > nat ) ).
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,
range2:
!>[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_Oreflp,type,
reflp:
!>[A: $tType] : ( ( A > A > $o ) > $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_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
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_Onormalization__semidom__class_Onormalize,type,
normal6383669964737779283malize:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
unit_f5069060285200089521factor:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Series_Osuminf,type,
suminf:
!>[A: $tType] : ( ( nat > A ) > A ) ).
thf(sy_c_Series_Osummable,type,
summable:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Series_Osums,type,
sums:
!>[A: $tType] : ( ( nat > A ) > A > $o ) ).
thf(sy_c_Set_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_Oimage,type,
image2:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_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_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
thf(sy_c_String_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_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
sum_case_sum:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
thf(sy_c_Topological__Spaces_Ocontinuous,type,
topolo3448309680560233919inuous:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Topological__Spaces_Ocontinuous__on,type,
topolo81223032696312382ous_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Topological__Spaces_Omonoseq,type,
topological_monoseq:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
topolo1002775350975398744n_open:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
topolo3827282254853284352ce_Lim:
!>[F: $tType,A: $tType] : ( ( filter @ F ) > ( F > A ) > A ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
topolo174197925503356063within:
!>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_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,
log2: real > real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Opowr,type,
powr:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Transcendental_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_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_Ortranclp,type,
transitive_rtranclp:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Otranclp,type,
transitive_tranclp:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_Typerep_Otyperep_OTyperep,type,
typerep2: literal > ( list @ typerep ) > typerep ).
thf(sy_c_Typerep_Otyperep_Osize__typerep,type,
size_typerep: typerep > nat ).
thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
vEBT_Node: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_Ocase__VEBT,type,
vEBT_case_VEBT:
!>[A: $tType] : ( ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A ) > ( $o > $o > A ) > vEBT_VEBT > A ) ).
thf(sy_c_VEBT__Definitions_OVEBT_Orec__VEBT,type,
vEBT_rec_VEBT:
!>[A: $tType] : ( ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A ) > ( $o > $o > A ) > vEBT_VEBT > A ) ).
thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
vEBT_V312737461966249ad_rel: ( product_prod @ vEBT_VEBT @ extended_enat ) > ( product_prod @ vEBT_VEBT @ extended_enat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
vEBT_V5917875025757280293ildren: nat > ( list @ vEBT_VEBT ) > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
vEBT_VEBT_valid_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_Oset__vebt,type,
vEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(sy_c_VEBT__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__comp__shift,type,
vEBT_V6923181176774028177_shift:
!>[A: $tType] : ( ( A > A > $o ) > ( option @ A ) > ( option @ A ) > $o ) ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__comp__shift__rel,type,
vEBT_V4810408830578336424ft_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) > $o ) ).
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_Ooption__shift__rel,type,
vEBT_V459564278314245337ft_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) > ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) > $o ) ).
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_Vector__Spaces_Olinear,type,
vector_linear:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > B ) > ( A > C > C ) > ( B > C ) > $o ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_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_Wellfounded_OwfP,type,
wfP:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Wfrec_Ocut,type,
cut:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ ( product_prod @ A @ A ) ) > A > A > B ) ).
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_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_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_a____,type,
a: $o ).
thf(sy_v_b____,type,
b: $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 (7542)
thf(fact_0_case4_I9_J,axiom,
ord_less_eq @ nat @ mi @ ma ).
% case4(9)
thf(fact_1_option_Oinject,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( ( some @ A @ X2 )
= ( some @ A @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_2_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_3_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_4_prod__decode__aux_Ocases,axiom,
! [X: product_prod @ nat @ nat] :
~ ! [K: nat,M: nat] :
( X
!= ( product_Pair @ nat @ nat @ K @ M ) ) ).
% prod_decode_aux.cases
thf(fact_5_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_6_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_7_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A5: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_8_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_9_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_10_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_11_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_12_max__in__set__def,axiom,
( vEBT_VEBT_max_in_set
= ( ^ [Xs: set @ nat,X4: nat] :
( ( member @ nat @ X4 @ Xs )
& ! [Y4: nat] :
( ( member @ nat @ Y4 @ Xs )
=> ( ord_less_eq @ nat @ Y4 @ X4 ) ) ) ) ) ).
% max_in_set_def
thf(fact_13_min__in__set__def,axiom,
( vEBT_VEBT_min_in_set
= ( ^ [Xs: set @ nat,X4: nat] :
( ( member @ nat @ X4 @ Xs )
& ! [Y4: nat] :
( ( member @ nat @ Y4 @ Xs )
=> ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) ) ).
% min_in_set_def
thf(fact_14_lesseq__shift,axiom,
( ( ord_less_eq @ nat )
= ( ^ [X4: nat,Y4: nat] : ( vEBT_VEBT_lesseq @ ( some @ nat @ X4 ) @ ( some @ nat @ Y4 ) ) ) ) ).
% lesseq_shift
thf(fact_15_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ 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 ) ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct7
thf(fact_16_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ 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 ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct6
thf(fact_17_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : ( P2 @ ( 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 ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct5
thf(fact_18_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B4: B,C2: C,D2: D] : ( P2 @ ( 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 ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct4
thf(fact_19_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B4: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_20_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_21_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_22_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_23_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_24_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_25_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B2: C] :
( ( produc5280177257484947105e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( C3 @ A3 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_26_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ A ),As: A > B] :
! [I: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_27_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F2: nat > A > A,A5: nat,B4: nat,Acc: A] :
( X
!= ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A5 @ ( product_Pair @ nat @ A @ B4 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_28_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_29_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_30_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N2 )
=> ( ord_less_eq @ A @ ( F3 @ N ) @ ( F3 @ N2 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_31_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N3 ) ) @ ( F3 @ N3 ) )
=> ( ( ord_less_eq @ nat @ N @ N2 )
=> ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( F3 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_32_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_33_le__trans,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less_eq @ nat @ J2 @ K2 )
=> ( ord_less_eq @ nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_34__C1_C_I2_J,axiom,
( deg
= ( suc @ ( zero_zero @ nat ) ) ) ).
% "1"(2)
thf(fact_35_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_36_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_37_prod__encode__eq,axiom,
! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_encode @ X )
= ( nat_prod_encode @ Y ) )
= ( X = Y ) ) ).
% prod_encode_eq
thf(fact_38_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_39_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_40_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_41_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_42_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( P2 @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M2 @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P2: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
( ! [X3: A] :
( ( F3 @ X3 )
= ( G3 @ X3 ) )
=> ( F3 = G3 ) ) ).
% ext
thf(fact_47_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_48_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_49_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_50_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_51_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_52_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_53_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_54_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_55_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% order_antisym_conv
thf(fact_56_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_le_cases
thf(fact_57_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 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_58_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 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_59_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_linear
thf(fact_60_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_eq_refl
thf(fact_61_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 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_62_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 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_63_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A7: A,B5: A] :
( ( ord_less_eq @ A @ A7 @ B5 )
& ( ord_less_eq @ A @ B5 @ A7 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_64_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_65_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G3 ) ) ) ).
% le_funI
thf(fact_66_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funE
thf(fact_67_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funD
thf(fact_68_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_69_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_70_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_71_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A7: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A7 )
& ( ord_less_eq @ A @ A7 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_72_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A3: A,B2: A] :
( ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( P2 @ A5 @ B4 ) )
=> ( ! [A5: A,B4: A] :
( ( P2 @ B4 @ A5 )
=> ( P2 @ A5 @ B4 ) )
=> ( P2 @ A3 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_73_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% order_trans
thf(fact_74_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_75_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% order_antisym
thf(fact_76_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_77_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_78_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_79_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_80_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_81_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M5: nat] :
( ( P2 @ X )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq @ nat @ X3 @ M5 ) )
=> ~ ! [M: nat] :
( ( P2 @ M )
=> ~ ! [X5: nat] :
( ( P2 @ X5 )
=> ( ord_less_eq @ nat @ X5 @ M ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_82_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq @ nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_83_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_84_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_85_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_86_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_87_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% bot_nat_0.extremum
thf(fact_88_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_89_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_90_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_91_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_92_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_93_Zero__not__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_94_Zero__neq__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_95_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_96_zero__induct,axiom,
! [P2: nat > $o,K2: nat] :
( ( P2 @ K2 )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_97_diff__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [X3: nat] : ( P2 @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y3: nat] : ( P2 @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P2 @ X3 @ Y3 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_98_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_99_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( ( zero_zero @ nat )
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_100_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_101_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_102_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_103_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_104_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_105_case4_I12_J,axiom,
vEBT_invar_vebt @ sa @ deg ).
% case4(12)
thf(fact_106_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_107_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_108_vebt__buildup_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ( ( X
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va: nat] :
( X
!= ( suc @ ( suc @ Va ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_109_dependent__nat__choice,axiom,
! [A: $tType,P2: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_1: A] : ( P2 @ ( zero_zero @ nat ) @ X_1 )
=> ( ! [X3: A,N3: nat] :
( ( P2 @ N3 @ X3 )
=> ? [Y6: A] :
( ( P2 @ ( suc @ N3 ) @ Y6 )
& ( Q @ N3 @ X3 @ Y6 ) ) )
=> ? [F2: nat > A] :
! [N4: nat] :
( ( P2 @ N4 @ ( F2 @ N4 ) )
& ( Q @ N4 @ ( F2 @ N4 ) @ ( F2 @ ( suc @ N4 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_110_exists__least__lemma,axiom,
! [P2: nat > $o] :
( ~ ( P2 @ ( zero_zero @ nat ) )
=> ( ? [X_1: nat] : ( P2 @ X_1 )
=> ? [N3: nat] :
( ~ ( P2 @ N3 )
& ( P2 @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_111_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_112_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_113_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q2: A,R3: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R3 ) )
= ( R3
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_114_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_115__C1_C_I1_J,axiom,
( sa
= ( vEBT_Leaf @ a @ b ) ) ).
% "1"(1)
thf(fact_116_valid__tree__deg__neq__0,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_tree_deg_neq_0
thf(fact_117_valid__0__not,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_0_not
thf(fact_118_Leaf__0__not,axiom,
! [A3: $o,B2: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B2 ) @ ( zero_zero @ nat ) ) ).
% Leaf_0_not
thf(fact_119_prod__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( ( nat_prod_decode @ X )
= ( nat_prod_decode @ Y ) )
= ( X = Y ) ) ).
% prod_decode_eq
thf(fact_120_insert_H__pres__valid,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( vEBT_invar_vebt @ ( vEBT_VEBT_insert @ T2 @ X ) @ N ) ) ).
% insert'_pres_valid
thf(fact_121_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_122_mult__0__right,axiom,
! [M2: nat] :
( ( times_times @ nat @ M2 @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_123_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_124_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_125_prod__decode__inverse,axiom,
! [N: nat] :
( ( nat_prod_encode @ ( nat_prod_decode @ N ) )
= N ) ).
% prod_decode_inverse
thf(fact_126_prod__encode__inverse,axiom,
! [X: product_prod @ nat @ nat] :
( ( nat_prod_decode @ ( nat_prod_encode @ X ) )
= X ) ).
% prod_encode_inverse
thf(fact_127_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_128_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_129_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_130_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A7: A,B5: A] : ( times_times @ A @ B5 @ A7 ) ) ) ) ).
% mult.commute
thf(fact_131_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_132_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_133_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X: A,Y: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y ) )
=> ( X != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_134_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_135_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_136_mult__le__mono2,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ I2 ) @ ( times_times @ nat @ K2 @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_137_mult__le__mono1,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K2 ) @ ( times_times @ nat @ J2 @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_138_mult__le__mono,axiom,
! [I2: nat,J2: nat,K2: nat,L: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less_eq @ nat @ K2 @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K2 ) @ ( times_times @ nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_139_le__square,axiom,
! [M2: nat] : ( ord_less_eq @ nat @ M2 @ ( times_times @ nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_140_le__cube,axiom,
! [M2: nat] : ( ord_less_eq @ nat @ M2 @ ( times_times @ nat @ M2 @ ( times_times @ nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_141_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_142_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_143_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_144_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ( times_times @ A @ A3 @ C3 )
= ( times_times @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_145_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ( times_times @ A @ C3 @ A3 )
= ( times_times @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_146_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_147_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_148_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_149_mul__shift,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( times_times @ nat @ X @ Y )
= Z3 )
= ( ( vEBT_VEBT_mul @ ( some @ nat @ X ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z3 ) ) ) ).
% mul_shift
thf(fact_150_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_151_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_152_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_153_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_154_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_155_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_156_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
( ( ( vEBT_Leaf @ X21 @ X22 )
= ( vEBT_Leaf @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% VEBT.inject(2)
thf(fact_157_mul__def,axiom,
( vEBT_VEBT_mul
= ( vEBT_V2048590022279873568_shift @ nat @ ( times_times @ nat ) ) ) ).
% mul_def
thf(fact_158_VEBT_Osize_I4_J,axiom,
! [X21: $o,X22: $o] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size(4)
thf(fact_159_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_160_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_161_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_162_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C3 @ A3 )
= ( times_times @ A @ C3 @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_163_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C3 )
= ( times_times @ A @ B2 @ C3 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_164_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_165_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_166_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_167_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_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_177_valid__eq1,axiom,
! [T2: vEBT_VEBT,D3: nat] :
( ( vEBT_invar_vebt @ T2 @ D3 )
=> ( vEBT_VEBT_valid @ T2 @ D3 ) ) ).
% valid_eq1
thf(fact_178_valid__eq2,axiom,
! [T2: vEBT_VEBT,D3: nat] :
( ( vEBT_VEBT_valid @ T2 @ D3 )
=> ( vEBT_invar_vebt @ T2 @ D3 ) ) ).
% valid_eq2
thf(fact_179_deg1Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
= ( ? [A7: $o,B5: $o] :
( T2
= ( vEBT_Leaf @ A7 @ B5 ) ) ) ) ).
% deg1Leaf
thf(fact_180_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_181_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_182_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times @ nat @ K2 @ M2 )
= ( times_times @ nat @ K2 @ N ) )
= ( ( K2
= ( zero_zero @ nat ) )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_183_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_184_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X22: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size_gen(2)
thf(fact_185_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
=> ? [Info: option @ ( product_prod @ nat @ nat ),TreeList: list @ vEBT_VEBT,S: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info @ ( suc @ ( suc @ N ) ) @ TreeList @ S ) ) ) ).
% deg_SUcn_Node
thf(fact_186_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_187_deg__deg__n,axiom,
! [Info2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
=> ( Deg = N ) ) ).
% deg_deg_n
thf(fact_188_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_189_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_190_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% mult_1
thf(fact_191_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.right_neutral
thf(fact_192_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_193_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_194_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_195_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_196_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_197_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_198_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_199_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_200_case4_I3_J,axiom,
vEBT_invar_vebt @ summary2 @ m ).
% case4(3)
thf(fact_201_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B2: A] :
( ( C3
= ( times_times @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_202_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,A3: A] :
( ( ( times_times @ A @ C3 @ A3 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_203_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B2: A] :
( ( C3
= ( times_times @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_204_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A,C3: A] :
( ( ( times_times @ A @ A3 @ C3 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_205_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_206_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_207_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ord_less @ nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_208_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_209_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_210_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_211_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_212_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_213_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_214_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_215_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_216_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_12: A] : ( ord_less @ A @ X @ X_12 ) ) ).
% gt_ex
thf(fact_217_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z: A] :
( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% dense
thf(fact_218_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_219_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_220_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_221_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_222_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X3 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A3 ) ) ) ).
% less_induct
thf(fact_223_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_224_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_225_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_226_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_227_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P3: A > $o] :
? [X6: A] : ( P3 @ X6 ) )
= ( ^ [P4: A > $o] :
? [N5: A] :
( ( P4 @ N5 )
& ! [M6: A] :
( ( ord_less @ A @ M6 @ N5 )
=> ~ ( P4 @ M6 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_228_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A3: A,B2: A] :
( ! [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
=> ( P2 @ A5 @ B4 ) )
=> ( ! [A5: A] : ( P2 @ A5 @ A5 )
=> ( ! [A5: A,B4: A] :
( ( P2 @ B4 @ A5 )
=> ( P2 @ A5 @ B4 ) )
=> ( P2 @ A3 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_229_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_230_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_231_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_232_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_233_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_234_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_235_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_236_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_237_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less @ nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_238_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P2: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y6: A] :
( ( ord_less @ B @ ( F3 @ Y6 ) @ ( F3 @ X3 ) )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A3 ) ) ) ).
% measure_induct
thf(fact_239_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_240_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_241_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_242_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_243_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P2: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y6: A] :
( ( ord_less @ B @ ( F3 @ Y6 ) @ ( F3 @ X3 ) )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_244_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE
thf(fact_245_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_asym
thf(fact_246_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neq_iff
thf(fact_247_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_248_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% order_less_trans
thf(fact_249_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 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_250_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 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ B @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_251_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_252_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 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_253_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 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% order_less_subst2
thf(fact_254_infinite__descent__measure,axiom,
! [A: $tType,P2: A > $o,V: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P2 @ X3 )
=> ? [Y6: A] :
( ( ord_less @ nat @ ( V @ Y6 ) @ ( V @ X3 ) )
& ~ ( P2 @ Y6 ) ) )
=> ( P2 @ X ) ) ).
% infinite_descent_measure
thf(fact_255_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_not_sym
thf(fact_256_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P2: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P2 ) ) ) ).
% order_less_imp_triv
thf(fact_257_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_less_linear
thf(fact_258_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% order_less_imp_not_eq
thf(fact_259_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% order_less_imp_not_eq2
thf(fact_260_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_imp_not_less
thf(fact_261_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_262_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_263_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_264_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_265_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_266_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ M2 ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_267_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N @ N2 )
=> ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ N2 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_268_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M2 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M2 @ N ) ) ) ) ) ).
% less_1_mult
thf(fact_269_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,D3: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D3 )
= ( D3
= ( one_one @ nat ) ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_270_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P2 @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_271_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_272_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_le_less_linear
thf(fact_273_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 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_274_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 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_275_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 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C3 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_276_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 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_277_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% order_less_le_trans
thf(fact_278_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% order_le_less_trans
thf(fact_279_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_280_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_281_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_less_imp_le
thf(fact_282_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_not_less
thf(fact_283_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_not_le
thf(fact_284_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ) ).
% order_less_le
thf(fact_285_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ) ).
% order_le_less
thf(fact_286_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_287_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_288_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A7: A] :
( ( ord_less_eq @ A @ B5 @ A7 )
& ~ ( ord_less_eq @ A @ A7 @ B5 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_289_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_290_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_291_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A7: A] :
( ( ord_less_eq @ A @ B5 @ A7 )
& ( A7 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_292_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A7: A] :
( ( ord_less @ A @ B5 @ A7 )
| ( A7 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_293_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z3 ) ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% dense_le_bounded
thf(fact_294_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less @ A @ Z3 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z3 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% dense_ge_bounded
thf(fact_295_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A7: A,B5: A] :
( ( ord_less_eq @ A @ A7 @ B5 )
& ~ ( ord_less_eq @ A @ B5 @ A7 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_296_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_297_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_298_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A7: A,B5: A] :
( ( ord_less_eq @ A @ A7 @ B5 )
& ( A7 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_299_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B5: A] :
( ( ord_less @ A @ A7 @ B5 )
| ( A7 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_300_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_301_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ~ ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_302_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z3: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ).
% dense_le
thf(fact_303_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z3: A,Y: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ).
% dense_ge
thf(fact_304_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_305_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_306_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_307_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_308_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_309_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_310_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_311_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_312_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_313_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_314_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_315_strict__inc__induct,axiom,
! [I2: nat,J2: nat,P2: nat > $o] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_316_less__Suc__induct,axiom,
! [I2: nat,J2: nat,P2: nat > nat > $o] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ K )
=> ( ( P2 @ I3 @ J3 )
=> ( ( P2 @ J3 @ K )
=> ( P2 @ I3 @ K ) ) ) ) )
=> ( P2 @ I2 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_317_less__trans__Suc,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ord_less @ nat @ ( suc @ I2 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_318_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_319_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_320_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M2 )
= ( ? [M7: nat] :
( ( M2
= ( suc @ M7 ) )
& ( ord_less @ nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_321_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I: nat] :
( ( ord_less @ nat @ I @ ( suc @ N ) )
=> ( P2 @ I ) ) )
= ( ( P2 @ N )
& ! [I: nat] :
( ( ord_less @ nat @ I @ N )
=> ( P2 @ I ) ) ) ) ).
% All_less_Suc
thf(fact_322_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M2 @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_323_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_324_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I: nat] :
( ( ord_less @ nat @ I @ ( suc @ N ) )
& ( P2 @ I ) ) )
= ( ( P2 @ N )
| ? [I: nat] :
( ( ord_less @ nat @ I @ N )
& ( P2 @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_325_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_326_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_327_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_328_Suc__lessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ K2 )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_329_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_330_Nat_OlessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ K2 )
=> ( ( K2
!= ( suc @ I2 ) )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_331_VEBT_Odistinct_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X21 @ X22 ) ) ).
% VEBT.distinct(1)
thf(fact_332_VEBT_Oexhaust,axiom,
! [Y: vEBT_VEBT] :
( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: list @ vEBT_VEBT,X142: vEBT_VEBT] :
( Y
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X222: $o] :
( Y
!= ( vEBT_Leaf @ X212 @ X222 ) ) ) ).
% VEBT.exhaust
thf(fact_333_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_334_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_335_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_336_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_337_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_338_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_339_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_340_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_341_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_342_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P2: A > $o,X: A] :
( ! [X3: A] :
( ( ( V @ X3 )
= ( zero_zero @ nat ) )
=> ( P2 @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
=> ( ~ ( P2 @ X3 )
=> ? [Y6: A] :
( ( ord_less @ nat @ ( V @ Y6 ) @ ( V @ X3 ) )
& ~ ( P2 @ Y6 ) ) ) )
=> ( P2 @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_343_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_344_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_345_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I2: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ord_less @ nat @ ( F3 @ I3 ) @ ( F3 @ J3 ) ) )
=> ( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ nat @ ( F3 @ I2 ) @ ( F3 @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_346_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_347_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_348_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M6: nat,N5: nat] :
( ( ord_less @ nat @ M6 @ N5 )
| ( M6 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_349_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_350_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M6: nat,N5: nat] :
( ( ord_less_eq @ nat @ M6 @ N5 )
& ( M6 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_351_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_352_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_353_nat__mult__eq__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ( times_times @ nat @ K2 @ M2 )
= ( times_times @ nat @ K2 @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_354_nat__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_355_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_356_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_357_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_358_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_359_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_360_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_361_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_362_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_363_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_364_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( times_times @ A @ A3 @ A3 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_365_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_366_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_367_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_368_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_369_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_370_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_371_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_372_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_373_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_374_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_375_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_376_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_377_mult__less__cancel__left__disj,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 @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_378_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_379_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_380_mult__less__cancel__right__disj,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 @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_381_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_382_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_383_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_384_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_385_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_386_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ? [J: nat] :
( ( M2
= ( suc @ J ) )
& ( ord_less @ nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_387_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_388_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I: nat] :
( ( ord_less @ nat @ I @ ( suc @ N ) )
=> ( P2 @ I ) ) )
= ( ( P2 @ ( zero_zero @ nat ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ N )
=> ( P2 @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_389_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_390_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I: nat] :
( ( ord_less @ nat @ I @ ( suc @ N ) )
& ( P2 @ I ) ) )
= ( ( P2 @ ( zero_zero @ nat ) )
| ? [I: nat] :
( ( ord_less @ nat @ I @ N )
& ( P2 @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_391_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_392_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N5: nat] : ( ord_less_eq @ nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_393_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_394_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_395_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_396_inc__induct,axiom,
! [I2: nat,J2: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( P2 @ J2 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I2 @ N3 )
=> ( ( ord_less @ nat @ N3 @ J2 )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% inc_induct
thf(fact_397_dec__induct,axiom,
! [I2: nat,J2: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( P2 @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I2 @ N3 )
=> ( ( ord_less @ nat @ N3 @ J2 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J2 ) ) ) ) ).
% dec_induct
thf(fact_398_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_399_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_400_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less_eq @ nat @ K @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_401_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_402_mult__less__mono1,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less @ nat @ ( times_times @ nat @ I2 @ K2 ) @ ( times_times @ nat @ J2 @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_403_mult__less__mono2,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less @ nat @ ( times_times @ nat @ K2 @ I2 ) @ ( times_times @ nat @ K2 @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_404_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_405_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_406_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_407_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_408_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_409_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_410_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_411_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_412_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_413_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_414_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_415_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_416_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_417_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_418_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_419_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_420_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X3 ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) @ Ux ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList @ S ) @ X3 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_421_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_422_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_423_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_424_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_425_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less @ nat @ K @ N )
& ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ K )
=> ~ ( P2 @ I4 ) )
& ( P2 @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_426_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_427_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_428_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_429_greater__shift,axiom,
( ( ord_less @ nat )
= ( ^ [Y4: nat,X4: nat] : ( vEBT_VEBT_greater @ ( some @ nat @ X4 ) @ ( some @ nat @ Y4 ) ) ) ) ).
% greater_shift
thf(fact_430_less__shift,axiom,
( ( ord_less @ nat )
= ( ^ [X4: nat,Y4: nat] : ( vEBT_VEBT_less @ ( some @ nat @ X4 ) @ ( some @ nat @ Y4 ) ) ) ) ).
% less_shift
thf(fact_431_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_432_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less @ A @ Z @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z @ X ) @ Y ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_433_case4_I2_J,axiom,
! [S2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ S2 @ m )
=> ( ( ( vEBT_VEBT_set_vebt @ summary2 )
= ( vEBT_VEBT_set_vebt @ S2 ) )
=> ( S2 = summary2 ) ) ) ).
% case4(2)
thf(fact_434_maxt__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) )
=> ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X ) ) ) ).
% maxt_corr
thf(fact_435_maxt__sound,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) ) ) ) ).
% maxt_sound
thf(fact_436_mint__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X ) )
=> ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X ) ) ) ).
% mint_corr
thf(fact_437_mint__sound,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X )
=> ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X ) ) ) ) ).
% mint_sound
thf(fact_438_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ ( times_times @ A @ Z3 @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_439_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y @ Z3 ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_440_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_441_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_442_case4_I5_J,axiom,
m = na ).
% case4(5)
thf(fact_443_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_444_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_445_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_446_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y @ Z3 ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% mult_less_iff1
thf(fact_447_vebt__maxt_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
= ( some @ nat @ Ma ) ) ).
% vebt_maxt.simps(3)
thf(fact_448_vebt__mint_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
= ( some @ nat @ Mi ) ) ).
% vebt_mint.simps(3)
thf(fact_449_maxt__corr__help,axiom,
! [T2: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ Maxi ) )
=> ( ( vEBT_vebt_member @ T2 @ X )
=> ( ord_less_eq @ nat @ X @ Maxi ) ) ) ) ).
% maxt_corr_help
thf(fact_450_mint__corr__help,axiom,
! [T2: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ Mini ) )
=> ( ( vEBT_vebt_member @ T2 @ X )
=> ( ord_less_eq @ nat @ Mini @ X ) ) ) ) ).
% mint_corr_help
thf(fact_451_buildup__nothing__in__leaf,axiom,
! [N: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_452_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_453_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_454_aa,axiom,
ord_less_eq @ ( set @ nat ) @ ( insert @ nat @ mi @ ( insert @ 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_455_VEBT__internal_Oinsert_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X3 ) )
=> ~ ! [Info: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ X3 ) ) ) ).
% VEBT_internal.insert'.cases
thf(fact_456_member__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ T2 @ X )
= ( member @ nat @ X @ ( vEBT_set_vebt @ T2 ) ) ) ) ).
% member_correct
thf(fact_457_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_458_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% buildup_gives_empty
thf(fact_459_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_460_pred__member,axiom,
! [T2: vEBT_VEBT,X: nat,Y: nat] :
( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X @ Y )
= ( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ Y @ X )
& ! [Z4: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z4 )
& ( ord_less @ nat @ Z4 @ X ) )
=> ( ord_less_eq @ nat @ Z4 @ Y ) ) ) ) ).
% pred_member
thf(fact_461_succ__member,axiom,
! [T2: vEBT_VEBT,X: nat,Y: nat] :
( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X @ Y )
= ( ( vEBT_vebt_member @ T2 @ Y )
& ( ord_less @ nat @ X @ Y )
& ! [Z4: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z4 )
& ( ord_less @ nat @ X @ Z4 ) )
=> ( ord_less_eq @ nat @ Y @ Z4 ) ) ) ) ).
% succ_member
thf(fact_462_case4_I6_J,axiom,
( deg
= ( plus_plus @ nat @ na @ m ) ) ).
% case4(6)
thf(fact_463_case4_I1_J,axiom,
! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ treeList2 ) )
=> ( ( vEBT_invar_vebt @ X5 @ na )
& ! [Xa: vEBT_VEBT] :
( ( vEBT_invar_vebt @ Xa @ na )
=> ( ( ( vEBT_VEBT_set_vebt @ X5 )
= ( vEBT_VEBT_set_vebt @ Xa ) )
=> ( Xa = X5 ) ) ) ) ) ).
% case4(1)
thf(fact_464_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_465_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_466_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_467_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_468_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_469_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_470_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw2 ) @ Ux2 ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_471_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A3: $o,B2: $o,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A3 @ B2 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A3 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B2 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_472_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X )
| ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% member_valid_both_member_options
thf(fact_473_vebt__member_Osimps_I4_J,axiom,
! [V3: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X ) ).
% vebt_member.simps(4)
thf(fact_474_vebt__member_Osimps_I1_J,axiom,
! [A3: $o,B2: $o,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A3 @ B2 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A3 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B2 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_475_vebt__member_Osimps_I3_J,axiom,
! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz ) @ X ) ).
% vebt_member.simps(3)
thf(fact_476_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_477_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_478_buildup__nothing__in__min__max,axiom,
! [N: nat,X: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_min_max
thf(fact_479_dele__member__cont__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T2 @ X ) @ Y )
= ( ( X != Y )
& ( vEBT_vebt_member @ T2 @ Y ) ) ) ) ).
% dele_member_cont_corr
thf(fact_480_VEBT__internal_Omembermima_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,Uw: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw ) )
=> ( ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT,Uz2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Uz2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb2 ) @ X3 ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ X3 ) )
=> ~ ! [V2: nat,TreeList: list @ vEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd ) @ X3 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_481_case4_I8_J,axiom,
( ( mi = ma )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ treeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ).
% case4(8)
thf(fact_482_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P2: A > $o,K2: A,F3: A > nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less @ nat @ ( F3 @ Y3 ) @ B2 ) )
=> ? [X3: A] :
( ( P2 @ X3 )
& ! [Y6: A] :
( ( P2 @ Y6 )
=> ( ord_less_eq @ nat @ ( F3 @ Y6 ) @ ( F3 @ X3 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_483_even__odd__cases,axiom,
! [X: nat] :
( ! [N3: nat] :
( X
!= ( plus_plus @ nat @ N3 @ N3 ) )
=> ~ ! [N3: nat] :
( X
!= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% even_odd_cases
thf(fact_484_delete__pres__valid,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T2 @ X ) @ N ) ) ).
% delete_pres_valid
thf(fact_485_maxbmo,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) )
=> ( vEBT_V8194947554948674370ptions @ T2 @ X ) ) ).
% maxbmo
thf(fact_486_dele__bmo__cont__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T2 @ X ) @ Y )
= ( ( X != Y )
& ( vEBT_V8194947554948674370ptions @ T2 @ Y ) ) ) ) ).
% dele_bmo_cont_corr
thf(fact_487_both__member__options__equiv__member,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
= ( vEBT_vebt_member @ T2 @ X ) ) ) ).
% both_member_options_equiv_member
thf(fact_488_valid__member__both__member__options,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
=> ( vEBT_vebt_member @ T2 @ X ) ) ) ).
% valid_member_both_member_options
thf(fact_489_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_490_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_491_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_492_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
=> ( ( Mi = Ma )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
& ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_493_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_494_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_495_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_496_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_497_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_498_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_499_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_500_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_501_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_502_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_503_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add_0
thf(fact_504_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_505_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_506_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_507_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% Nat.add_0_right
thf(fact_508_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_509_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_510_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_511_not__Some__eq,axiom,
! [A: $tType,X: option @ A] :
( ( ! [Y4: A] :
( X
!= ( some @ A @ Y4 ) ) )
= ( X
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_512_not__None__eq,axiom,
! [A: $tType,X: option @ A] :
( ( X
!= ( none @ A ) )
= ( ? [Y4: A] :
( X
= ( some @ A @ Y4 ) ) ) ) ).
% not_None_eq
thf(fact_513_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_514_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_515_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_516_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_517_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_518_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_519_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_520_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_521_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_522_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_523_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_524_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_525_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_526_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_527_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_528_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_529_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_530_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A7: A,B5: A] : ( plus_plus @ A @ B5 @ A7 ) ) ) ) ).
% add.commute
thf(fact_531_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_532_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_533_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_534_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: A,K2: A,B2: A,A3: A] :
( ( B6
= ( plus_plus @ A @ K2 @ B2 ) )
=> ( ( plus_plus @ A @ A3 @ B6 )
= ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_535_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_536_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J2: A,K2: A,L: A] :
( ( ( I2 = J2 )
& ( K2 = L ) )
=> ( ( plus_plus @ A @ I2 @ K2 )
= ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_537_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_538_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_539_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_540_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Uz ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_541_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J2: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J2 )
& ( K2 = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_542_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J2: A,K2: A,L: A] :
( ( ( I2 = J2 )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_543_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J2: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J2 )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_544_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_545_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_546_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_547_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_548_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B5: A] :
? [C4: A] :
( B5
= ( plus_plus @ A @ A7 @ C4 ) ) ) ) ) ).
% le_iff_add
thf(fact_549_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_550_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_551_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_552_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_553_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_554_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_555_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_556_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_557_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_558_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_559_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J2: A,K2: A,L: A] :
( ( ( ord_less @ A @ I2 @ J2 )
& ( K2 = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_560_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J2: A,K2: A,L: A] :
( ( ( I2 = J2 )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_561_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J2: A,K2: A,L: A] :
( ( ( ord_less @ A @ I2 @ J2 )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_562_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_563_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_564_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_565_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_566_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_567_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_568_combine__options__cases,axiom,
! [A: $tType,B: $tType,X: option @ A,P2: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
( ( ( X
= ( none @ A ) )
=> ( P2 @ X @ Y ) )
=> ( ( ( Y
= ( none @ B ) )
=> ( P2 @ X @ Y ) )
=> ( ! [A5: A,B4: B] :
( ( X
= ( some @ A @ A5 ) )
=> ( ( Y
= ( some @ B @ B4 ) )
=> ( P2 @ X @ Y ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_569_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P3: ( option @ A ) > $o] :
! [X6: option @ A] : ( P3 @ X6 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
& ! [X4: A] : ( P4 @ ( some @ A @ X4 ) ) ) ) ) ).
% split_option_all
thf(fact_570_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P3: ( option @ A ) > $o] :
? [X6: option @ A] : ( P3 @ X6 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
| ? [X4: A] : ( P4 @ ( some @ A @ X4 ) ) ) ) ) ).
% split_option_ex
thf(fact_571_option_Oexhaust,axiom,
! [A: $tType,Y: option @ A] :
( ( Y
!= ( none @ A ) )
=> ~ ! [X23: A] :
( Y
!= ( some @ A @ X23 ) ) ) ).
% option.exhaust
thf(fact_572_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X2: A] :
( ( Option
= ( some @ A @ X2 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_573_option_Odistinct_I1_J,axiom,
! [A: $tType,X2: A] :
( ( none @ A )
!= ( some @ A @ X2 ) ) ).
% option.distinct(1)
thf(fact_574_VEBT__internal_Ooption__shift_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu2: A > A > A,Uv2: option @ A] :
( X
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) )
=> ( ! [Uw: A > A > A,V2: A] :
( X
!= ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V2 ) @ ( none @ A ) ) ) )
=> ~ ! [F2: A > A > A,A5: A,B4: A] :
( X
!= ( 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_575_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] :
( ! [Uu2: A > A > $o,Uv2: option @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) )
=> ( ! [Uw: A > A > $o,V2: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V2 ) @ ( none @ A ) ) ) )
=> ~ ! [F2: A > A > $o,X3: A,Y3: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X3 ) @ ( some @ A @ Y3 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_576_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_577_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_578_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_579_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_580_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_581_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_582_trans__less__add2,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_583_trans__less__add1,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_584_add__less__mono1,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J2 @ K2 ) ) ) ).
% add_less_mono1
thf(fact_585_not__add__less2,axiom,
! [J2: nat,I2: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J2 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_586_not__add__less1,axiom,
! [I2: nat,J2: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J2 ) @ I2 ) ).
% not_add_less1
thf(fact_587_add__less__mono,axiom,
! [I2: nat,J2: nat,K2: nat,L: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ K2 @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_588_add__lessD1,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J2 ) @ K2 )
=> ( ord_less @ nat @ I2 @ K2 ) ) ).
% add_lessD1
thf(fact_589_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_590_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) ) ).
% le_add1
thf(fact_591_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M2 @ N ) ) ).
% le_add2
thf(fact_592_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_593_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_594_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq @ nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus @ nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_595_add__le__mono,axiom,
! [I2: nat,J2: nat,K2: nat,L: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less_eq @ nat @ K2 @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_596_add__le__mono1,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J2 @ K2 ) ) ) ).
% add_le_mono1
thf(fact_597_trans__le__add1,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_598_trans__le__add2,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_599_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M6: nat,N5: nat] :
? [K3: nat] :
( N5
= ( plus_plus @ nat @ M6 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_600_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_601_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_602_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J2: nat,K2: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ K2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I2 @ J2 ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_603_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P2: A > $o,K2: A,F3: A > nat,N: nat] :
( ( P2 @ K2 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ? [Y6: A] :
( ( P2 @ Y6 )
& ~ ( ord_less_eq @ nat @ ( F3 @ Y6 ) @ ( F3 @ X3 ) ) ) )
=> ? [Y3: A] :
( ( P2 @ Y3 )
& ~ ( ord_less @ nat @ ( F3 @ Y3 ) @ ( plus_plus @ nat @ ( F3 @ K2 ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_604_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,Uw2: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw2 ) ).
% VEBT_internal.membermima.simps(1)
thf(fact_605_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_606_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_607_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_608_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_609_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_610_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_611_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_612_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_613_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_614_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_615_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J2: A,K2: A,L: A] :
( ( ( ord_less @ A @ I2 @ J2 )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_616_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J2: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J2 )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_617_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_618_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_619_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_620_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_621_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_622_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_623_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_624_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_625_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_626_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_627_less__add__Suc1,axiom,
! [I2: nat,M2: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ I2 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_628_less__add__Suc2,axiom,
! [I2: nat,M2: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ M2 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_629_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M6: nat,N5: nat] :
? [K3: nat] :
( N5
= ( suc @ ( plus_plus @ nat @ M6 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_630_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_631_less__imp__add__positive,axiom,
! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ? [K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ( plus_plus @ nat @ I2 @ K )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_632_mono__nat__linear__lb,axiom,
! [F3: nat > nat,M2: nat,K2: nat] :
( ! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( ord_less @ nat @ ( F3 @ M ) @ ( F3 @ N3 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F3 @ M2 ) @ K2 ) @ ( F3 @ ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_633_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_634_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_635_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_636_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_637_VEBT__internal_OminNull_Ocases,axiom,
! [X: vEBT_VEBT] :
( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.cases
thf(fact_638_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [A: $tType,Uw2: A > A > A,V3: A] :
( ( vEBT_V2048590022279873568_shift @ A @ Uw2 @ ( some @ A @ V3 ) @ ( none @ A ) )
= ( none @ A ) ) ).
% VEBT_internal.option_shift.simps(2)
thf(fact_639_VEBT__internal_Ooption__shift_Oelims,axiom,
! [A: $tType,X: A > A > A,Xa2: option @ A,Xb: option @ A,Y: option @ A] :
( ( ( vEBT_V2048590022279873568_shift @ A @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2
= ( none @ A ) )
=> ( Y
!= ( none @ A ) ) )
=> ( ( ? [V2: A] :
( Xa2
= ( some @ A @ V2 ) )
=> ( ( Xb
= ( none @ A ) )
=> ( Y
!= ( none @ A ) ) ) )
=> ~ ! [A5: A] :
( ( Xa2
= ( some @ A @ A5 ) )
=> ! [B4: A] :
( ( Xb
= ( some @ A @ B4 ) )
=> ( Y
!= ( some @ A @ ( X @ A5 @ B4 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_640_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y @ E2 ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_epsilon
thf(fact_641_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_642_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_643_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_644_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_645_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_646_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_647_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_648_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_649_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A7: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A7 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_650_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_651_ex__has__least__nat,axiom,
! [A: $tType,P2: A > $o,K2: A,M2: A > nat] :
( ( P2 @ K2 )
=> ? [X3: A] :
( ( P2 @ X3 )
& ! [Y6: A] :
( ( P2 @ Y6 )
=> ( ord_less_eq @ nat @ ( M2 @ X3 ) @ ( M2 @ Y6 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_652_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_653_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A3: A,Y: A,U: A,V3: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V3 )
=> ( ( ( plus_plus @ A @ U @ V3 )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V3 @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_654_vebt__mint_Oelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( some @ nat @ Mi2 ) ) ) ) ) ) ).
% vebt_mint.elims
thf(fact_655_vebt__maxt_Oelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y
= ( none @ nat ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( some @ nat @ Ma2 ) ) ) ) ) ) ).
% vebt_maxt.elims
thf(fact_656_vebt__mint_Ocases,axiom,
! [X: vEBT_VEBT] :
( ! [A5: $o,B4: $o] :
( X
!= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ).
% vebt_mint.cases
thf(fact_657_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A3: A,Y: A,U: A,V3: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ( ord_less @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V3 )
=> ( ( ( plus_plus @ A @ U @ V3 )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V3 @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_658_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va3: list @ vEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb ) @ X )
= ( ( X = Mi )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_659_vebt__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X3 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) @ X3 ) )
=> ( ! [V2: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ X3 ) )
=> ( ! [V2: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_660_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_661_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_662_geqmaxNone,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Ma @ X )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( none @ nat ) ) ) ) ).
% geqmaxNone
thf(fact_663_vebt__pred_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) )
=> ( ! [A5: $o,Uw: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A5: $o,B4: $o,Va: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ Va ) ) ) )
=> ( ! [Uy: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va2 ) @ Vb2 ) )
=> ( ! [V2: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Vf ) )
=> ( ! [V2: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Vj ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% vebt_pred.cases
thf(fact_664_vebt__succ_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,B4: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ ( zero_zero @ nat ) ) )
=> ( ! [Uv2: $o,Uw: $o,N3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw ) @ ( suc @ N3 ) ) )
=> ( ! [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) @ Va2 ) )
=> ( ! [V2: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) @ Ve ) )
=> ( ! [V2: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Vi ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).
% vebt_succ.cases
thf(fact_665_vebt__delete_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( zero_zero @ nat ) ) )
=> ( ! [A5: $o,B4: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A5: $o,B4: $o,N3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ ( suc @ ( suc @ N3 ) ) ) )
=> ( ! [Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList @ Summary2 ) @ Uu2 ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst: list @ vEBT_VEBT,Smry: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) @ X3 ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr: list @ vEBT_VEBT,Sm: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% vebt_delete.cases
thf(fact_666_vebt__insert_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A5: $o,B4: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X3 ) )
=> ( ! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) @ X3 ) )
=> ( ! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) @ X3 ) )
=> ( ! [V2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% vebt_insert.cases
thf(fact_667_succ__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T2 ) @ X @ Sx ) ) ) ).
% succ_correct
thf(fact_668_pred__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat @ Sx ) )
= ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T2 ) @ X @ Sx ) ) ) ).
% pred_correct
thf(fact_669_pred__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Px: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat @ Px ) )
= ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X @ Px ) ) ) ).
% pred_corr
thf(fact_670_succ__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X )
= ( some @ nat @ Sx ) )
= ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X @ Sx ) ) ) ).
% succ_corr
thf(fact_671_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_672_vebt__succ_Osimps_I2_J,axiom,
! [Uv: $o,Uw2: $o,N: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw2 ) @ ( suc @ N ) )
= ( none @ nat ) ) ).
% vebt_succ.simps(2)
thf(fact_673_vebt__pred_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o] :
( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ).
% vebt_pred.simps(1)
thf(fact_674_vebt__succ_Osimps_I4_J,axiom,
! [V3: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vc @ Vd2 ) @ Ve2 )
= ( none @ nat ) ) ).
% vebt_succ.simps(4)
thf(fact_675_vebt__pred_Osimps_I5_J,axiom,
! [V3: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Vf2 )
= ( none @ nat ) ) ).
% vebt_pred.simps(5)
thf(fact_676_vebt__pred_Osimps_I6_J,axiom,
! [V3: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 )
= ( none @ nat ) ) ).
% vebt_pred.simps(6)
thf(fact_677_vebt__succ_Osimps_I5_J,axiom,
! [V3: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 )
= ( none @ nat ) ) ).
% vebt_succ.simps(5)
thf(fact_678_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_679_vebt__succ_Osimps_I1_J,axiom,
! [B2: $o,Uu: $o] :
( ( B2
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B2 ) @ ( zero_zero @ nat ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B2 ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ) ) ).
% vebt_succ.simps(1)
thf(fact_680_vebt__pred_Osimps_I3_J,axiom,
! [B2: $o,A3: $o,Va3: nat] :
( ( B2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A3
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_pred.simps(3)
thf(fact_681_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_682_vebt__delete_Osimps_I1_J,axiom,
! [A3: $o,B2: $o] :
( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A3 @ B2 ) @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ B2 ) ) ).
% vebt_delete.simps(1)
thf(fact_683_is__pred__in__set__def,axiom,
( vEBT_is_pred_in_set
= ( ^ [Xs: set @ nat,X4: nat,Y4: nat] :
( ( member @ nat @ Y4 @ Xs )
& ( ord_less @ nat @ Y4 @ X4 )
& ! [Z4: nat] :
( ( member @ nat @ Z4 @ Xs )
=> ( ( ord_less @ nat @ Z4 @ X4 )
=> ( ord_less_eq @ nat @ Z4 @ Y4 ) ) ) ) ) ) ).
% is_pred_in_set_def
thf(fact_684_is__succ__in__set__def,axiom,
( vEBT_is_succ_in_set
= ( ^ [Xs: set @ nat,X4: nat,Y4: nat] :
( ( member @ nat @ Y4 @ Xs )
& ( ord_less @ nat @ X4 @ Y4 )
& ! [Z4: nat] :
( ( member @ nat @ Z4 @ Xs )
=> ( ( ord_less @ nat @ X4 @ Z4 )
=> ( ord_less_eq @ nat @ Y4 @ Z4 ) ) ) ) ) ) ).
% is_succ_in_set_def
thf(fact_685_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_686_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_687_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_688_vebt__delete_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) ) ).
% vebt_delete.simps(5)
thf(fact_689_vebt__delete_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) ) ).
% vebt_delete.simps(6)
thf(fact_690_add__def,axiom,
( vEBT_VEBT_add
= ( vEBT_V2048590022279873568_shift @ nat @ ( plus_plus @ nat ) ) ) ).
% add_def
thf(fact_691_delete__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct
thf(fact_692_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_693_add__shift,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( plus_plus @ nat @ X @ Y )
= Z3 )
= ( ( vEBT_VEBT_add @ ( some @ nat @ X ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z3 ) ) ) ).
% add_shift
thf(fact_694_delete__correct_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% delete_correct'
thf(fact_695_obtain__set__succ,axiom,
! [X: nat,Z3: nat,A6: set @ nat,B6: set @ nat] :
( ( ord_less @ nat @ X @ Z3 )
=> ( ( vEBT_VEBT_max_in_set @ A6 @ Z3 )
=> ( ( finite_finite2 @ nat @ B6 )
=> ( ( A6 = B6 )
=> ? [X_12: nat] : ( vEBT_is_succ_in_set @ A6 @ X @ X_12 ) ) ) ) ) ).
% obtain_set_succ
thf(fact_696_obtain__set__pred,axiom,
! [Z3: nat,X: nat,A6: set @ nat] :
( ( ord_less @ nat @ Z3 @ X )
=> ( ( vEBT_VEBT_min_in_set @ A6 @ Z3 )
=> ( ( finite_finite2 @ nat @ A6 )
=> ? [X_12: nat] : ( vEBT_is_pred_in_set @ A6 @ X @ X_12 ) ) ) ) ).
% obtain_set_pred
thf(fact_697_length__pos__if__in__set,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_698_option_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X2: A] :
( ( size_option @ A @ X @ ( some @ A @ X2 ) )
= ( plus_plus @ nat @ ( X @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_699_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R3: A,A3: A,B2: A,C3: A,D3: A] :
( ( R3
!= ( zero_zero @ A ) )
=> ( ( ( A3 = B2 )
& ( C3 != D3 ) )
=> ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R3 @ C3 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R3 @ D3 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_700_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_701_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 )
=> ~ ? [X5: nat] :
( ( member @ nat @ X5 @ Xs2 )
& ( ord_less @ nat @ X5 @ A3 ) ) ) ) ).
% pred_none_empty
thf(fact_702_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 )
=> ~ ? [X5: nat] :
( ( member @ nat @ X5 @ Xs2 )
& ( ord_less @ nat @ A3 @ X5 ) ) ) ) ).
% succ_none_empty
thf(fact_703_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_704_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_0_right
thf(fact_705_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_706_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_zero
thf(fact_707_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_708_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_709_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_710_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_711_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_712_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_713_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_714_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_715_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_716_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_717_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_718_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_719_diff__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(9)
thf(fact_720_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_721_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs3: list @ A] :
( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_722_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_723_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_724_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_725_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_726_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_727_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_728_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A7: A,B5: A] :
( ( minus_minus @ A @ A7 @ B5 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_729_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_730_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_731_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_732_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_733_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% left_diff_distrib
thf(fact_734_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% right_diff_distrib
thf(fact_735_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( times_times @ A @ ( minus_minus @ A @ B2 @ C3 ) @ A3 )
= ( minus_minus @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C3 @ A3 ) ) ) ) ).
% left_diff_distrib'
thf(fact_736_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% right_diff_distrib'
thf(fact_737_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_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_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_746_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N6: set @ nat] :
? [M6: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ N6 )
=> ( ord_less @ nat @ X4 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_747_bounded__nat__set__is__finite,axiom,
! [N7: set @ nat,N: nat] :
( ! [X3: nat] :
( ( member @ nat @ X3 @ N7 )
=> ( ord_less @ nat @ X3 @ N ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% bounded_nat_set_is_finite
thf(fact_748_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N6: set @ nat] :
? [M6: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ N6 )
=> ( ord_less_eq @ nat @ X4 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_749_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A7 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_750_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A7: A,B5: A] : ( ord_less @ A @ ( minus_minus @ A @ A7 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_751_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: A,K2: A,N: A,J2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J2 @ K2 ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J2 @ K2 ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K2 ) @ J2 ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_752_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: A,K2: A,N: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ N )
=> ( ord_less_eq @ A @ I2 @ ( minus_minus @ A @ N @ K2 ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_753_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_754_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_755_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_756_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_757_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_758_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_759_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_760_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_761_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_762_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_763_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_764_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_765_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_766_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_767_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_768_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X: A,Y: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ A @ X @ Y ) ) ) ) ).
% square_diff_square_factored
thf(fact_769_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_770_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_771_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_772_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_773_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_774_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_775_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_776_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S3: set @ B,P2: ( set @ B ) > $o,F3: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( P2 @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S4: set @ B] :
( ( finite_finite2 @ B @ S4 )
=> ( ! [Y6: B] :
( ( member @ B @ Y6 @ S4 )
=> ( ord_less_eq @ A @ ( F3 @ Y6 ) @ ( F3 @ X3 ) ) )
=> ( ( P2 @ S4 )
=> ( P2 @ ( insert @ B @ X3 @ S4 ) ) ) ) )
=> ( P2 @ S3 ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_777_length__induct,axiom,
! [A: $tType,P2: ( 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 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs2 ) ) ).
% length_induct
thf(fact_778_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_779_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W2: A,Y: A,X: A,Z3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W2 @ Y ) @ ( times_times @ A @ X @ Z3 ) )
= ( plus_plus @ A @ ( times_times @ A @ W2 @ Z3 ) @ ( times_times @ A @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z3 ) ) ) ) ).
% crossproduct_eq
thf(fact_780_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_781_option_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_option @ A @ X @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_782_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ord_less_eq @ A @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_783_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ord_less_eq @ A @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_784_mult__diff__mult,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [X: A,Y: A,A3: A,B2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ Y ) @ ( times_times @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ X @ ( minus_minus @ A @ Y @ B2 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X @ A3 ) @ B2 ) ) ) ) ).
% mult_diff_mult
thf(fact_785_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( minus_minus @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% diff_shunt_var
thf(fact_786_minNullmin,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ T2 )
=> ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) ) ) ).
% minNullmin
thf(fact_787_minminNull,axiom,
! [T2: vEBT_VEBT] :
( ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) )
=> ( vEBT_VEBT_minNull @ T2 ) ) ).
% minminNull
thf(fact_788_vebt__maxt_Opelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ 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 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxt.pelims
thf(fact_789_vebt__mint_Opelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ X )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ 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 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_mint.pelims
thf(fact_790_infinite__nat__iff__unbounded__le,axiom,
! [S3: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S3 ) )
= ( ! [M6: nat] :
? [N5: nat] :
( ( ord_less_eq @ nat @ M6 @ N5 )
& ( member @ nat @ N5 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_791_inthall,axiom,
! [A: $tType,Xs2: list @ A,P2: A > $o,N: nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( P2 @ X3 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P2 @ ( nth @ A @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_792_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_793_min__Null__member,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ~ ( vEBT_vebt_member @ T2 @ X ) ) ).
% min_Null_member
thf(fact_794_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_795_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_796_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_797_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ M2 )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_798_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_799_diff__diff__left,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ K2 )
= ( minus_minus @ nat @ I2 @ ( plus_plus @ nat @ J2 @ K2 ) ) ) ).
% diff_diff_left
thf(fact_800_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_801_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_802_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_803_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J2: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J2 )
=> ( ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J2 @ K2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J2 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_804_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J2: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J2 )
=> ( ( plus_plus @ nat @ ( minus_minus @ nat @ J2 @ K2 ) @ I2 )
= ( minus_minus @ nat @ ( plus_plus @ nat @ J2 @ I2 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_805_Nat_Odiff__diff__right,axiom,
! [K2: nat,J2: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J2 )
=> ( ( minus_minus @ nat @ I2 @ ( minus_minus @ nat @ J2 @ K2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_806_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_807_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_808_diff__Suc__diff__eq1,axiom,
! [K2: nat,J2: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J2 )
=> ( ( minus_minus @ nat @ I2 @ ( suc @ ( minus_minus @ nat @ J2 @ K2 ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_809_diff__Suc__diff__eq2,axiom,
! [K2: nat,J2: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J2 )
=> ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J2 @ K2 ) ) @ I2 )
= ( minus_minus @ nat @ ( suc @ J2 ) @ ( plus_plus @ nat @ K2 @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_810_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_811_zero__induct__lemma,axiom,
! [P2: nat > $o,K2: nat,I2: nat] :
( ( P2 @ K2 )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus @ nat @ K2 @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_812_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% minus_nat.diff_0
thf(fact_813_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_814_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_815_less__imp__diff__less,axiom,
! [J2: nat,K2: nat,N: nat] :
( ( ord_less @ nat @ J2 @ K2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J2 @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_816_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_817_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_818_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_819_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_820_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_821_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_822_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_823_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_824_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_825_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_826_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_827_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_828_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_829_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_830_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_831_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_832_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z2: list @ A] : Y5 = Z2 )
= ( ^ [Xs: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I )
= ( nth @ A @ Ys3 @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_833_Skolem__list__nth,axiom,
! [A: $tType,K2: nat,P2: nat > A > $o] :
( ( ! [I: nat] :
( ( ord_less @ nat @ I @ K2 )
=> ? [X7: A] : ( P2 @ I @ X7 ) ) )
= ( ? [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= K2 )
& ! [I: nat] :
( ( ord_less @ nat @ I @ K2 )
=> ( P2 @ I @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_834_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_835_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_836_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_837_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_838_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_839_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_840_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_841_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_842_finite__maxlen,axiom,
! [A: $tType,M5: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M5 )
=> ? [N3: nat] :
! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ M5 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_843_less__diff__conv,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ ( minus_minus @ nat @ J2 @ K2 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ J2 ) ) ).
% less_diff_conv
thf(fact_844_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_845_le__diff__conv,axiom,
! [J2: nat,K2: nat,I2: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J2 @ K2 ) @ I2 )
= ( ord_less_eq @ nat @ J2 @ ( plus_plus @ nat @ I2 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_846_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J2: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J2 )
=> ( ( ord_less_eq @ nat @ I2 @ ( minus_minus @ nat @ J2 @ K2 ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_847_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J2: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J2 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J2 ) @ K2 )
= ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J2 @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_848_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J2: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J2 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ J2 @ I2 ) @ K2 )
= ( plus_plus @ nat @ ( minus_minus @ nat @ J2 @ K2 ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_849_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ( minus_minus @ nat @ J2 @ I2 )
= K2 )
= ( J2
= ( plus_plus @ nat @ K2 @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_850_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_851_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod @ nat @ nat,Va3: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va3 @ Vb @ Vc ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_852_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_853_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs2: list @ A,P2: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth @ A @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_854_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
= ( ? [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( nth @ A @ Xs2 @ I )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_855_all__nth__imp__all__set,axiom,
! [A: $tType,Xs2: list @ A,P2: A > $o,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P2 @ ( nth @ A @ Xs2 @ I3 ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_856_all__set__conv__all__nth,axiom,
! [A: $tType,Xs2: list @ A,P2: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( P2 @ X4 ) ) )
= ( ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P2 @ ( nth @ A @ Xs2 @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_857_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_858_nat__diff__split,axiom,
! [P2: nat > $o,A3: nat,B2: nat] :
( ( P2 @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( ( ( ord_less @ nat @ A3 @ B2 )
=> ( P2 @ ( zero_zero @ nat ) ) )
& ! [D4: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D4 ) )
=> ( P2 @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_859_nat__diff__split__asm,axiom,
! [P2: nat > $o,A3: nat,B2: nat] :
( ( P2 @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( ~ ( ( ( ord_less @ nat @ A3 @ B2 )
& ~ ( P2 @ ( zero_zero @ nat ) ) )
| ? [D4: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D4 ) )
& ~ ( P2 @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_860_less__diff__conv2,axiom,
! [K2: nat,J2: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J2 )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ J2 @ K2 ) @ I2 )
= ( ord_less @ nat @ J2 @ ( plus_plus @ nat @ I2 @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_861_nat__eq__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J2 @ I2 )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 )
= ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ U ) @ M2 )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_862_nat__eq__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 )
= ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_863_nat__le__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J2 @ I2 )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_864_nat__le__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_865_nat__diff__add__eq1,axiom,
! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J2 @ I2 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_866_nat__diff__add__eq2,axiom,
! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( minus_minus @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_867_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_868_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_869_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_870_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_871_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M6: nat,N5: nat] :
( if @ nat
@ ( M6
= ( zero_zero @ nat ) )
@ N5
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M6 @ ( one_one @ nat ) ) @ N5 ) ) ) ) ) ).
% add_eq_if
thf(fact_872_nat__less__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J2 @ I2 )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_873_nat__less__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_874_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M6: nat,N5: nat] :
( if @ nat
@ ( M6
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N5 @ ( times_times @ nat @ ( minus_minus @ nat @ M6 @ ( one_one @ nat ) ) @ N5 ) ) ) ) ) ).
% mult_eq_if
thf(fact_875_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y )
=> ( ( ? [Uv2: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y )
=> ( ( ? [Uu2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> Y )
=> ( ( ? [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
=> ~ Y )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> Y ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_876_field__lbound__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D1: A,D22: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
=> ? [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
& ( ord_less @ A @ E2 @ D1 )
& ( ord_less @ A @ E2 @ D22 ) ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_877_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ( ord_less_eq @ A @ X3 @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ord_less_eq @ A @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_878_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ( ord_less_eq @ A @ A3 @ X3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ord_less_eq @ A @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_879_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ~ Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
=> ( ! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
=> ( Y
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( 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_880_Suc__if__eq,axiom,
! [A: $tType,F3: nat > A,H: nat > A,G3: A,N: nat] :
( ! [N3: nat] :
( ( F3 @ ( suc @ N3 ) )
= ( H @ N3 ) )
=> ( ( ( 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_881_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_882_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
=> ( ! [Uu2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va2: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( 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_883_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_884_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_885_Euclid__induct,axiom,
! [P2: nat > nat > $o,A3: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( P2 @ A5 @ B4 )
= ( P2 @ B4 @ A5 ) )
=> ( ! [A5: nat] : ( P2 @ A5 @ ( zero_zero @ nat ) )
=> ( ! [A5: nat,B4: nat] :
( ( P2 @ A5 @ B4 )
=> ( P2 @ A5 @ ( plus_plus @ nat @ A5 @ B4 ) ) )
=> ( P2 @ A3 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_886_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( P2 @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ K @ I4 )
=> ( P2 @ I4 ) )
=> ( P2 @ K ) ) )
=> ( P2 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_887_length__mul__elem,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),N: nat] :
( ! [X3: list @ A] :
( ( member @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ X3 )
= N ) )
=> ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs2 ) )
= ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) @ N ) ) ) ).
% length_mul_elem
thf(fact_888_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_889_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_890_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_891_diff__commute,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J2 ) @ K2 )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ K2 ) @ J2 ) ) ).
% diff_commute
thf(fact_892_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_893_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M6: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M6 @ K3 ) @ ( product_Pair @ nat @ nat @ M6 @ ( minus_minus @ nat @ K3 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M6 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_894_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_895_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_896_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_897_Gcd__remove0__nat,axiom,
! [M5: set @ nat] :
( ( finite_finite2 @ nat @ M5 )
=> ( ( gcd_Gcd @ nat @ M5 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M5 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_898_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P2: A > $o,D5: A,Q: A > $o] :
( ! [X3: A,K: A] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K @ D5 ) ) ) )
=> ( ! [X3: A,K: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K @ D5 ) ) ) )
=> ! [X5: A,K4: A] :
( ( ( P2 @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K4 @ D5 ) ) )
| ( Q @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K4 @ D5 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_899_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P2: A > $o,D5: A,Q: A > $o] :
( ! [X3: A,K: A] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K @ D5 ) ) ) )
=> ( ! [X3: A,K: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K @ D5 ) ) ) )
=> ! [X5: A,K4: A] :
( ( ( P2 @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K4 @ D5 ) ) )
& ( Q @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K4 @ D5 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_900_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% verit_sum_simplify
thf(fact_901_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_902_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 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_903_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_904_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(2)
thf(fact_905_prod__decode__def,axiom,
( nat_prod_decode
= ( nat_prod_decode_aux @ ( zero_zero @ nat ) ) ) ).
% prod_decode_def
thf(fact_906_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_907_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% pinf(6)
thf(fact_908_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% pinf(8)
thf(fact_909_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% minf(6)
thf(fact_910_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ~ ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% minf(8)
thf(fact_911_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_912_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: A,B2: A,P2: A > $o] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( P2 @ A3 )
=> ( ~ ( P2 @ B2 )
=> ? [C2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
& ( ord_less_eq @ A @ C2 @ B2 )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less @ A @ X5 @ C2 ) )
=> ( P2 @ X5 ) )
& ! [D6: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less @ A @ X3 @ D6 ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq @ A @ D6 @ C2 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_913_vebt__insert_Osimps_I4_J,axiom,
! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ X ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_914_nth__zip,axiom,
! [A: $tType,B: $tType,I2: nat,Xs2: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) @ I2 )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ B @ Ys @ I2 ) ) ) ) ) ).
% nth_zip
thf(fact_915_find__Some__iff2,axiom,
! [A: $tType,X: A,P2: A > $o,Xs2: list @ A] :
( ( ( some @ A @ X )
= ( find @ A @ P2 @ Xs2 ) )
= ( ? [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P2 @ ( nth @ A @ Xs2 @ I ) )
& ( X
= ( nth @ A @ Xs2 @ I ) )
& ! [J: nat] :
( ( ord_less @ nat @ J @ I )
=> ~ ( P2 @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_916_find__Some__iff,axiom,
! [A: $tType,P2: A > $o,Xs2: list @ A,X: A] :
( ( ( find @ A @ P2 @ Xs2 )
= ( some @ A @ X ) )
= ( ? [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P2 @ ( nth @ A @ Xs2 @ I ) )
& ( X
= ( nth @ A @ Xs2 @ I ) )
& ! [J: nat] :
( ( ord_less @ nat @ J @ I )
=> ~ ( P2 @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_917_power_Opower__eq__if,axiom,
! [A: $tType] :
( ( power2 @ A )
= ( ^ [One: A,Times: A > A > A,P5: A,M6: nat] :
( if @ A
@ ( M6
= ( zero_zero @ nat ) )
@ One
@ ( Times @ P5 @ ( power2 @ A @ One @ Times @ P5 @ ( minus_minus @ nat @ M6 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% power.power_eq_if
thf(fact_918_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_919_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_920_length__rotate1,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_rotate1
thf(fact_921_nth__Cons__Suc,axiom,
! [A: $tType,X: A,Xs2: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth @ A @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_922_nth__Cons__0,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ ( zero_zero @ nat ) )
= X ) ).
% nth_Cons_0
thf(fact_923_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X: A,Xs2: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y @ Ys ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_924_enumerate__simps_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs2: list @ B] :
( ( enumerate @ B @ N @ ( cons @ B @ X @ Xs2 ) )
= ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X ) @ ( enumerate @ B @ ( suc @ N ) @ Xs2 ) ) ) ).
% enumerate_simps(2)
thf(fact_925_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,Xy: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs2 @ Ys )
= ( cons @ ( product_prod @ A @ B ) @ Xy @ Xys ) )
=> ~ ! [X3: A,Xs4: list @ A] :
( ( Xs2
= ( cons @ A @ X3 @ Xs4 ) )
=> ! [Y3: B,Ys4: list @ B] :
( ( Ys
= ( cons @ B @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_926_find_Osimps_I2_J,axiom,
! [A: $tType,P2: A > $o,X: A,Xs2: list @ A] :
( ( ( P2 @ X )
=> ( ( find @ A @ P2 @ ( cons @ A @ X @ Xs2 ) )
= ( some @ A @ X ) ) )
& ( ~ ( P2 @ X )
=> ( ( find @ A @ P2 @ ( cons @ A @ X @ Xs2 ) )
= ( find @ A @ P2 @ Xs2 ) ) ) ) ).
% find.simps(2)
thf(fact_927_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs2: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) )
=> ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_928_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,Xs2: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs2 ) ) ) ).
% set_zip_leftD
thf(fact_929_in__set__zipE,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs2: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) )
=> ~ ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ~ ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_930_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_931_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs2 ) )
= ( ? [Y4: A,Ys3: list @ A] :
( ( Xs2
= ( cons @ A @ Y4 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_932_length__Suc__conv,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N ) )
= ( ? [Y4: A,Ys3: list @ A] :
( ( Xs2
= ( cons @ A @ Y4 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_933_impossible__Cons,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs2
!= ( cons @ A @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_934_power_Opower_Opower__Suc,axiom,
! [A: $tType,One2: A,Times2: A > A > A,A3: A,N: nat] :
( ( power2 @ A @ One2 @ Times2 @ A3 @ ( suc @ N ) )
= ( Times2 @ A3 @ ( power2 @ A @ One2 @ Times2 @ A3 @ N ) ) ) ).
% power.power.power_Suc
thf(fact_935_power_Opower_Opower__0,axiom,
! [A: $tType,One2: A,Times2: A > A > A,A3: A] :
( ( power2 @ A @ One2 @ Times2 @ A3 @ ( zero_zero @ nat ) )
= One2 ) ).
% power.power.power_0
thf(fact_936_vebt__insert_Osimps_I2_J,axiom,
! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) @ X )
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S2 ) ) ).
% vebt_insert.simps(2)
thf(fact_937_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ~ ! [Y3: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_938_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 ) )
=> ~ ! [X3: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_939_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_940_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
! [A3: $o,B2: $o,X: nat] :
( ( vEBT_VEBT_insert @ ( vEBT_Leaf @ A3 @ B2 ) @ X )
= ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B2 ) @ X ) ) ).
% VEBT_internal.insert'.simps(1)
thf(fact_941_vebt__insert_Osimps_I3_J,axiom,
! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S2: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) @ X )
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S2 ) ) ).
% vebt_insert.simps(3)
thf(fact_942_vebt__insert_Osimps_I1_J,axiom,
! [X: nat,A3: $o,B2: $o] :
( ( ( X
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B2 ) @ X )
= ( vEBT_Leaf @ $true @ B2 ) ) )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B2 ) @ X )
= ( vEBT_Leaf @ A3 @ $true ) ) )
& ( ( X
!= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B2 ) @ X )
= ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ) ) ).
% vebt_insert.simps(1)
thf(fact_943_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X22 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size(4)
thf(fact_944_nth__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= X ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_945_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs2: list @ A,N: nat] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= X )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_946_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs2: list @ A,N: nat] :
( ( X != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_947_length__Cons,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_Cons
thf(fact_948_dbl__inc__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_inc_simps(2)
thf(fact_949_enumerate__Suc_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ ( infini527867602293511546merate @ A @ S3 @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N ) ) ) ).
% enumerate_Suc'
thf(fact_950_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_951_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 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite2 @ A @ A6 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_952_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_953_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_954_count__notin,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( count_list @ A @ Xs2 @ X )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_955_Cons__lenlex__iff,axiom,
! [A: $tType,M2: A,Ms: list @ A,N: A,Ns: list @ A,R3: 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 @ R3 ) )
= ( ( 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 ) @ R3 ) )
| ( ( M2 = N )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R3 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_956_power__shift,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( power_power @ nat @ X @ Y )
= Z3 )
= ( ( vEBT_VEBT_power @ ( some @ nat @ X ) @ ( some @ nat @ Y ) )
= ( some @ nat @ Z3 ) ) ) ).
% power_shift
thf(fact_957_local_Opower__def,axiom,
( vEBT_VEBT_power
= ( vEBT_V2048590022279873568_shift @ nat @ ( power_power @ nat ) ) ) ).
% local.power_def
thf(fact_958_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M2: nat] :
( ( ( power_power @ nat @ X @ M2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ( X
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_959_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_960_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_961_dbl__dec__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_dec_simps(3)
thf(fact_962_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_963_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_964_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_965_card_Oinfinite,axiom,
! [A: $tType,A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( finite_card @ A @ A6 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_966_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_967_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_968_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_969_card__insert__disjoint,axiom,
! [A: $tType,A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X @ A6 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
= ( suc @ ( finite_card @ A @ A6 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_970_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_971_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_972_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_973_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_974_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ Y )
= ( times_times @ A @ Y @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_975_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_976_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_977_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_978_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_979_lenlex__irreflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( lenlex @ A @ R3 ) ) ) ).
% lenlex_irreflexive
thf(fact_980_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_981_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_982_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_983_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B6: set @ A,A6: set @ B,R3: B > A > $o] :
( ( finite_finite2 @ A @ B6 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ A6 )
=> ? [B7: A] :
( ( member @ A @ B7 @ B6 )
& ( R3 @ A5 @ B7 ) ) )
=> ( ! [A1: B,A22: B,B4: A] :
( ( member @ B @ A1 @ A6 )
=> ( ( member @ B @ A22 @ A6 )
=> ( ( member @ A @ B4 @ B6 )
=> ( ( R3 @ A1 @ B4 )
=> ( ( R3 @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A6 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_984_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_985_card__insert__le,axiom,
! [A: $tType,A6: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ ( insert @ A @ X @ A6 ) ) ) ).
% card_insert_le
thf(fact_986_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_987_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_988_finite__enum__subset,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X8: set @ A,Y7: set @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( finite_card @ A @ X8 ) )
=> ( ( infini527867602293511546merate @ A @ X8 @ I3 )
= ( infini527867602293511546merate @ A @ Y7 @ I3 ) ) )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ X8 ) @ ( finite_card @ A @ Y7 ) )
=> ( ord_less_eq @ ( set @ A ) @ X8 @ Y7 ) ) ) ) ) ) ).
% finite_enum_subset
thf(fact_989_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_990_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_991_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_992_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_993_nat__power__less__imp__less,axiom,
! [I2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I2 )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I2 @ M2 ) @ ( power_power @ nat @ I2 @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_994_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_995_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_996_card__ge__0__finite,axiom,
! [A: $tType,A6: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A6 ) )
=> ( finite_finite2 @ A @ A6 ) ) ).
% card_ge_0_finite
thf(fact_997_card__insert__if,axiom,
! [A: $tType,A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( member @ A @ X @ A6 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
= ( finite_card @ A @ A6 ) ) )
& ( ~ ( member @ A @ X @ A6 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
= ( suc @ ( finite_card @ A @ A6 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_998_card__Suc__eq__finite,axiom,
! [A: $tType,A6: set @ A,K2: nat] :
( ( ( finite_card @ A @ A6 )
= ( suc @ K2 ) )
= ( ? [B5: A,B8: set @ A] :
( ( A6
= ( insert @ A @ B5 @ B8 ) )
& ~ ( member @ A @ B5 @ B8 )
& ( ( finite_card @ A @ B8 )
= K2 )
& ( finite_finite2 @ A @ B8 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_999_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_1000_card__mono,axiom,
! [A: $tType,B6: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B6 ) ) ) ) ).
% card_mono
thf(fact_1001_card__seteq,axiom,
! [A: $tType,B6: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B6 ) @ ( finite_card @ A @ A6 ) )
=> ( A6 = B6 ) ) ) ) ).
% card_seteq
thf(fact_1002_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F5: set @ A,C5: nat] :
( ! [G5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G5 @ F5 )
=> ( ( finite_finite2 @ A @ G5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G5 ) @ C5 ) ) )
=> ( ( finite_finite2 @ A @ F5 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F5 ) @ C5 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1003_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S3: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S3 ) )
=> ~ ! [T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 )
=> ( ( ( finite_card @ A @ T4 )
= N )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1004_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_1005_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_1006_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_1007_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_1008_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_1009_card__le__sym__Diff,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B6 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B6 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B6 @ A6 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1010_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_1011_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_1012_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_1013_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N7: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N7 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N7 ) ) ) ) ) ).
% power_increasing
thf(fact_1014_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_1015_lenlex__length,axiom,
! [A: $tType,Ms: list @ A,Ns: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R3 ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% lenlex_length
thf(fact_1016_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_1017_nat__one__le__power,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I2 )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I2 @ N ) ) ) ).
% nat_one_le_power
thf(fact_1018_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_1019_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_1020_card__1__singleton__iff,axiom,
! [A: $tType,A6: set @ A] :
( ( ( finite_card @ A @ A6 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X4: A] :
( A6
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1021_card__eq__SucD,axiom,
! [A: $tType,A6: set @ A,K2: nat] :
( ( ( finite_card @ A @ A6 )
= ( suc @ K2 ) )
=> ? [B4: A,B9: set @ A] :
( ( A6
= ( insert @ A @ B4 @ B9 ) )
& ~ ( member @ A @ B4 @ B9 )
& ( ( finite_card @ A @ B9 )
= K2 )
& ( ( K2
= ( zero_zero @ nat ) )
=> ( B9
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_1022_card__Suc__eq,axiom,
! [A: $tType,A6: set @ A,K2: nat] :
( ( ( finite_card @ A @ A6 )
= ( suc @ K2 ) )
= ( ? [B5: A,B8: set @ A] :
( ( A6
= ( insert @ A @ B5 @ B8 ) )
& ~ ( member @ A @ B5 @ B8 )
& ( ( finite_card @ A @ B8 )
= K2 )
& ( ( K2
= ( zero_zero @ nat ) )
=> ( B8
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1023_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 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ A6 )
=> ( X4 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1024_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A6: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A6 ) )
= ( ? [A7: A,B8: set @ A] :
( ( A6
= ( insert @ A @ A7 @ B8 ) )
& ~ ( member @ A @ A7 @ B8 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B8 ) )
& ( finite_finite2 @ A @ B8 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1025_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_1026_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_1027_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_1028_card__Diff1__le,axiom,
! [A: $tType,A6: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) ) ).
% card_Diff1_le
thf(fact_1029_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N7: nat,A3: A] :
( ( ord_less @ nat @ N @ N7 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N7 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1030_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N7: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N7 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N7 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1031_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_1032_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_1033_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_1034_diff__card__le__card__Diff,axiom,
! [A: $tType,B6: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B6 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B6 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1035_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_1036_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_1037_count__le__length,axiom,
! [A: $tType,Xs2: list @ A,X: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs2 @ X ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% count_le_length
thf(fact_1038_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_1039_card_Oremove,axiom,
! [A: $tType,A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( finite_card @ A @ A6 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1040_card_Oinsert__remove,axiom,
! [A: $tType,A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_1041_card__Suc__Diff1,axiom,
! [A: $tType,A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A6 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1042_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_1043_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P5: A,M6: nat] :
( if @ A
@ ( M6
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P5 @ ( power_power @ A @ P5 @ ( minus_minus @ nat @ M6 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1044_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_1045_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs2: list @ A,Y: A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R3 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lex @ A @ R3 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_1046_realpow__pos__nth__unique,axiom,
! [N: nat,A3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ( power_power @ real @ X3 @ N )
= A3 )
& ! [Y6: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y6 )
& ( ( power_power @ real @ Y6 @ N )
= A3 ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1047_realpow__pos__nth,axiom,
! [N: nat,A3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
& ( ( power_power @ real @ R4 @ N )
= A3 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1048_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 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ treeList ) )
=> ( vEBT_invar_vebt @ X5 @ na ) ) ) ).
% sprop1
thf(fact_1049_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R3 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [N5: nat] :
( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ N5 ) @ ( nth @ B @ Ys @ N5 ) ) @ R3 ) ) ) ) ).
% listrel_iff_nth
thf(fact_1050_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) )
= ( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X )
= ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_1051_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1052__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,Summary3: vEBT_VEBT,Info: option @ ( product_prod @ nat @ nat )] :
~ ( ( sa
= ( vEBT_Node @ Info @ deg @ TreeList3 @ Summary3 ) )
& ( deg
= ( plus_plus @ nat @ na @ m ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
& ( vEBT_invar_vebt @ Summary3 @ m )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ 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_1053_intind,axiom,
! [A: $tType,I2: nat,N: nat,P2: A > $o,X: A] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( P2 @ X )
=> ( P2 @ ( nth @ A @ ( replicate @ A @ N @ X ) @ I2 ) ) ) ) ).
% intind
thf(fact_1054_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_1055_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_1056_semiring__norm_I11_J,axiom,
! [M2: num] :
( ( times_times @ num @ M2 @ one2 )
= M2 ) ).
% semiring_norm(11)
thf(fact_1057_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times @ num @ one2 @ N )
= N ) ).
% semiring_norm(12)
thf(fact_1058_case4_I10_J,axiom,
ord_less @ nat @ ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ) ).
% case4(10)
thf(fact_1059_case4_I4_J,axiom,
( ( size_size @ ( list @ vEBT_VEBT ) @ treeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) ).
% case4(4)
thf(fact_1060_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_1061_a0,axiom,
ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ).
% a0
thf(fact_1062_case4_I7_J,axiom,
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList2 @ I4 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ summary2 @ I4 ) ) ) ).
% case4(7)
thf(fact_1063_member__bound,axiom,
! [Tree: vEBT_VEBT,X: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_1064_valid__pres__insert,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ T2 @ X ) @ N ) ) ) ).
% valid_pres_insert
thf(fact_1065_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_1066_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_1067_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_1068_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [V3: num,W2: num,Z3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Z3 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V3 @ W2 ) ) @ Z3 ) ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1069_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V3: num,W2: num,Z3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V3 ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Z3 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V3 @ W2 ) ) @ Z3 ) ) ) ).
% add_numeral_left
thf(fact_1070_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_1071_num__double,axiom,
! [N: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1072_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_1073_insert__simp__mima,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
| ( X = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_1074_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_1075_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less @ real @ ( zero_zero @ real ) @ ( times_times @ real @ X @ X ) ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% not_real_square_gt_zero
thf(fact_1076_valid__insert__both__member__options__add,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ X ) @ X ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_1077_valid__insert__both__member__options__pres,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ Y ) @ X ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_1078_helpypredd,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat @ Y ) )
=> ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpypredd
thf(fact_1079_helpyd,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X )
= ( some @ nat @ Y ) )
=> ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% helpyd
thf(fact_1080_post__member__pre__member,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T2 @ X ) @ Y )
=> ( ( vEBT_vebt_member @ T2 @ Y )
| ( X = Y ) ) ) ) ) ) ).
% post_member_pre_member
thf(fact_1081_replicate__eq__replicate,axiom,
! [A: $tType,M2: nat,X: A,N: nat,Y: A] :
( ( ( replicate @ A @ M2 @ X )
= ( replicate @ A @ N @ Y ) )
= ( ( M2 = N )
& ( ( M2
!= ( zero_zero @ nat ) )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_1082_length__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_1083_delt__out__of__range,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( 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 @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% delt_out_of_range
thf(fact_1084_del__single__cont,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
& ( X = 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 @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% del_single_cont
thf(fact_1085_set__n__deg__not__0,axiom,
! [TreeList2: list @ vEBT_VEBT,N: nat,M2: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_1086_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: 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 @ TreeList2 @ 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_1087_pred__max,axiom,
! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( some @ nat @ Ma ) ) ) ) ).
% pred_max
thf(fact_1088_succ__min,axiom,
! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( some @ nat @ Mi ) ) ) ) ).
% succ_min
thf(fact_1089_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H2: nat,L2: nat,D4: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D4 ) ) @ L2 ) ) ) ).
% bit_concat_def
thf(fact_1090_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V3: num,B2: A,C3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V3 ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V3 ) @ C3 ) ) ) ) ).
% distrib_left_numeral
thf(fact_1091_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A3: A,B2: A,V3: num] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V3 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V3 ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_1092_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_1093_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_1094_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A3: A,B2: A,V3: num] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V3 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V3 ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1095_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V3: num,B2: A,C3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V3 ) @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V3 ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V3 ) @ C3 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1096_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_1097_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_1098_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_1099_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_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_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_1111_in__set__replicate,axiom,
! [A: $tType,X: A,N: nat,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ Y ) ) )
= ( ( X = Y )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_1112_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P2: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
& ( P2 @ X4 ) ) )
= ( ( P2 @ A3 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_1113_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P2: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
=> ( P2 @ X4 ) ) )
= ( ( P2 @ A3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_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_1125_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_1126_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1127_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1128_set__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_1129_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X = Y ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1130_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_1131_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_1132_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1133_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: num,N: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) @ X ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_1134_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I2: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_1135_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_1136_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_1137_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_1138_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 )
& ? [Y4: A] :
( ( member @ A @ Y4 @ S3 )
& ( X4 != Y4 )
& ! [Z4: A] :
( ( member @ A @ Z4 @ S3 )
=> ( ( Z4 = X4 )
| ( Z4 = Y4 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_1139_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ! [Y6: real] :
? [N3: nat] : ( ord_less @ real @ Y6 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1140_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_1141_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_1142_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_1143_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_1144_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_1145_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_1146_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_1147_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq @ num @ X @ one2 )
= ( X = one2 ) ) ).
% le_num_One_iff
thf(fact_1148_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z3 )
= ( plus_plus @ A @ Z3 @ Z3 ) ) ) ).
% mult_2
thf(fact_1149_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z3: A] :
( ( times_times @ A @ Z3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z3 @ Z3 ) ) ) ).
% mult_2_right
thf(fact_1150_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_1151_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_1152_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_1153_power4__eq__xxxx,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A] :
( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X @ X ) @ X ) @ X ) ) ) ).
% power4_eq_xxxx
thf(fact_1154_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_1155_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_1156_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_1157_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_1158_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_sum
thf(fact_1159_sum__squares__bound,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_squares_bound
thf(fact_1160_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_diff
thf(fact_1161_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% power2_le_imp_le
thf(fact_1162_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( X = Y ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_1163_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_1164_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_1165_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_1166_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_1167_card__2__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X4: A,Y4: A] :
( ( S3
= ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X4 != Y4 ) ) ) ) ).
% card_2_iff
thf(fact_1168_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1169_Suc__nat__number__of__add,axiom,
! [V3: num,N: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V3 ) @ N ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V3 @ one2 ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1170_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N3: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% real_arch_simple
thf(fact_1171_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N3: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% reals_Archimedean2
thf(fact_1172_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X ) ) ) ) ).
% mult_of_nat_commute
thf(fact_1173_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W3: int,Z4: int] :
? [N5: nat] :
( Z4
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N5 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1174_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_1175_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_1176_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_1177_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_1178_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N5: nat,M6: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N5 ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M6 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_1179_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_1180_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A7: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1181_int__ops_I7_J,axiom,
! [A3: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A3 @ B2 ) )
= ( times_times @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_1182_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ X @ Y ) ) ) ) ).
% power2_less_imp_less
thf(fact_1183_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_1184_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_1185_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_1186_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_1187_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_1188_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_1189_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_1190_ex__power__ivl1,axiom,
! [B2: nat,K2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K2 )
=> ? [N3: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K2 )
& ( ord_less @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1191_ex__power__ivl2,axiom,
! [B2: nat,K2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K2 )
& ( ord_less_eq @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1192_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
! [Deg: nat,X: nat,Info2: option @ ( product_prod @ nat @ nat ),TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) @ X )
=> ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) )
& ( ~ ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) @ X )
=> ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% VEBT_internal.insert'.simps(2)
thf(fact_1193_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_1194_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_1195_VEBT__internal_Oinsert_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_insert @ X @ Xa2 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y
!= ( vEBT_vebt_insert @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) ) )
=> ~ ! [Info: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
=> ~ ( ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa2 )
=> ( Y
= ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) ) )
& ( ~ ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa2 )
=> ( Y
= ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.insert'.elims
thf(fact_1196_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_1197_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_1198_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_1199_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_1200_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_1201_of__nat__mono,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J2 ) ) ) ) ).
% of_nat_mono
thf(fact_1202_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_1203_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_1204_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_1205_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_1206_replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] :
( ( replicate @ A @ ( suc @ N ) @ X )
= ( cons @ A @ X @ ( replicate @ A @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_1207_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C3 )
=> ( ! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ X ) @ C3 ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1208_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_1209_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ~ ! [N3: nat] :
( ( K2
= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1210_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( K2
= ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1211_zmult__zless__mono2__lemma,axiom,
! [I2: int,J2: int,K2: nat] :
( ( ord_less @ int @ I2 @ J2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K2 ) @ I2 ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K2 ) @ J2 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1212_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_1213_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_1214_replicate__length__same,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( X3 = X ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ X )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_1215_replicate__eqI,axiom,
! [A: $tType,Xs2: list @ A,N: nat,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate @ A @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_1216_invar__vebt_Ointros_I2_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( 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 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_1217_invar__vebt_Ointros_I3_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( 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 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_1218_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R3 ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_1219_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1220_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_1221_zdiff__int__split,axiom,
! [P2: int > $o,X: nat,Y: nat] :
( ( P2 @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y ) ) )
= ( ( ( ord_less_eq @ nat @ Y @ X )
=> ( P2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y ) ) ) )
& ( ( ord_less @ nat @ X @ Y )
=> ( P2 @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_1222_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_1223_realpow__pos__nth2,axiom,
! [A3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
& ( ( power_power @ real @ R4 @ ( suc @ N ) )
= A3 ) ) ) ).
% realpow_pos_nth2
thf(fact_1224_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Y: B,Ys: list @ B,R3: 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 @ R3 ) )
=> ~ ! [X3: A,Xs3: list @ A] :
( ( Xs2
= ( cons @ A @ X3 @ Xs3 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_1225_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list @ A,Xs2: list @ B,R3: 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 @ R3 ) )
=> ~ ! [Y3: B,Ys5: list @ B] :
( ( Xs2
= ( cons @ B @ Y3 @ Ys5 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys5 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_1226_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R3: set @ ( product_prod @ A @ B ),Xs2: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R3 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R3 ) ) ) ) ).
% listrel.Cons
thf(fact_1227_measures__less,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) ) ) ).
% measures_less
thf(fact_1228_measures__lesseq,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_1229_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N ) @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_1230_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1231_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs2: list @ A,N: nat,Y: A] :
( ( ( cons @ A @ X @ Xs2 )
= ( replicate @ A @ N @ Y ) )
= ( ( X = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs2
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_1232_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R3: A,Q2: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R3 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q2 @ R3 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q2 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R3 @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R3 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q2 @ R3 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q2 ) @ R3 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_1233_mintlistlength,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: 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 @ TreeList2 @ 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_1234_insert__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( sup_sup @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) )
= ( vEBT_set_vebt @ ( vEBT_vebt_insert @ T2 @ X ) ) ) ) ) ).
% insert_correct
thf(fact_1235_insert__corr,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( sup_sup @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) )
= ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_insert @ T2 @ X ) ) ) ) ) ).
% insert_corr
thf(fact_1236_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_1237_nat__bit__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( P2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_1238_nat__induct2,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ( P2 @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct2
thf(fact_1239_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_1240_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_1241_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_1242_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_1243_bits__div__by__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_1244_bits__div__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_1245_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_1246_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_1247_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_1248_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ C3 )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% divide_cancel_right
thf(fact_1249_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ( divide_divide @ A @ C3 @ A3 )
= ( divide_divide @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% divide_cancel_left
thf(fact_1250_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_1251_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% times_divide_eq_right
thf(fact_1252_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ).
% divide_divide_eq_right
thf(fact_1253_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_1254_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( times_times @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C3 ) ) ) ).
% times_divide_eq_left
thf(fact_1255_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% div_by_1
thf(fact_1256_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I2 )
& ( ord_less_eq @ A @ I2 @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_1257_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_1258_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_1259_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ A3 )
= B2 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1260_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1261_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1262_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1263_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1264_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1265_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1266_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_1267_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_1268_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_1269_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B2 @ A3 ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_1270_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ( divide_divide @ A @ B2 @ A3 )
= ( one_one @ A ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_1271_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_1272_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_1273_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A3 @ B2 ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_1274_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_1275_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_1276_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_1277_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_1278_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_1279_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Icc_iff
thf(fact_1280_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
( ( set_or1337092689740270186AtMost @ A @ A3 @ A3 )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_1281_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B2 )
& ( B2 = C3 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_1282_nat__mult__div__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( K2
= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( zero_zero @ nat ) ) )
& ( ( K2
!= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( divide_divide @ nat @ M2 @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1283_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_1284_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_1285_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_1286_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1287_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1288_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1289_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1290_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_1291_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_1292_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_1293_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) )
= A3 )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1294_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1295_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_1296_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_1297_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1298_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1299_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_1300_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_1301_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_1302_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_1303_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_1304_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_1305_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_1306_decr__mult__lemma,axiom,
! [D3: int,P2: int > $o,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int] :
( ( P2 @ X3 )
=> ( P2 @ ( minus_minus @ int @ X3 @ D3 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ! [X5: int] :
( ( P2 @ X5 )
=> ( P2 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D3 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1307_minusinfinity,axiom,
! [D3: int,P1: int > $o,P2: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int,K: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K @ D3 ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_1308_plusinfinity,axiom,
! [D3: int,P6: int > $o,P2: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int,K: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K @ D3 ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [X_1: int] : ( P6 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_1309_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_1310_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times @ int @ K2 @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_1311_pos__zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M2 )
=> ( ( ( times_times @ int @ M2 @ N )
= ( one_one @ int ) )
= ( ( M2
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1312_zmult__zless__mono2,axiom,
! [I2: int,J2: int,K2: int] :
( ( ord_less @ int @ I2 @ J2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ord_less @ int @ ( times_times @ int @ K2 @ I2 ) @ ( times_times @ int @ K2 @ J2 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1313_unique__quotient__lemma__neg,axiom,
! [B2: int,Q4: int,R5: int,Q2: int,R3: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R5 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R3 )
=> ( ( ord_less @ int @ B2 @ R5 )
=> ( ord_less_eq @ int @ Q2 @ Q4 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_1314_unique__quotient__lemma,axiom,
! [B2: int,Q4: int,R5: int,Q2: int,R3: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R5 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R5 )
=> ( ( ord_less @ int @ R5 @ B2 )
=> ( ( ord_less @ int @ R3 @ B2 )
=> ( ord_less_eq @ int @ Q4 @ Q2 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_1315_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q2: int,R3: int,B3: int,Q4: int,R5: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R3 )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R5 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R5 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R3 @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ Q4 @ Q2 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_1316_zdiv__mono2__lemma,axiom,
! [B2: int,Q2: int,R3: int,B3: int,Q4: int,R5: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R3 )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R5 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R5 ) )
=> ( ( ord_less @ int @ R5 @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ Q2 @ Q4 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_1317_q__pos__lemma,axiom,
! [B3: int,Q4: int,R5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q4 ) @ R5 ) )
=> ( ( ord_less @ int @ R5 @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q4 ) ) ) ) ).
% q_pos_lemma
thf(fact_1318_incr__mult__lemma,axiom,
! [D3: int,P2: int > $o,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int] :
( ( P2 @ X3 )
=> ( P2 @ ( plus_plus @ int @ X3 @ D3 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ! [X5: int] :
( ( P2 @ X5 )
=> ( P2 @ ( plus_plus @ int @ X5 @ ( times_times @ int @ K2 @ D3 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1319_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_1320_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times @ int @ W2 @ ( plus_plus @ int @ Z1 @ Z22 ) )
= ( plus_plus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1321_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_1322_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1323_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times @ int @ W2 @ ( minus_minus @ int @ Z1 @ Z22 ) )
= ( minus_minus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1324_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_1325_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z3: A,W2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z3 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ W2 ) @ ( times_times @ A @ Y @ Z3 ) ) ) ) ).
% divide_divide_times_eq
thf(fact_1326_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z3: A,W2: A] :
( ( times_times @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z3 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y @ W2 ) ) ) ) ).
% times_divide_times_eq
thf(fact_1327_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_1328_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Icc
thf(fact_1329_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_1330_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_1331_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1332_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1333_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1334_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1335_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_1336_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_1337_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_1338_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_1339_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1340_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1341_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) )
& ( C3
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_1342_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_1343_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_pos_pos
thf(fact_1344_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_1345_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_1346_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_neg_neg
thf(fact_1347_all__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M6: nat] :
( ( ord_less_eq @ nat @ M6 @ N )
=> ( P2 @ M6 ) ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P2 @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_1348_ex__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M6: nat] :
( ( ord_less_eq @ nat @ M6 @ N )
& ( P2 @ M6 ) ) )
= ( ? [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P2 @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_1349_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z3: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y )
= ( divide_divide @ A @ W2 @ Z3 ) )
= ( ( times_times @ A @ X @ Z3 )
= ( times_times @ A @ W2 @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1350_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ( divide_divide @ A @ B2 @ C3 )
= A3 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_1351_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_1352_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A3 @ C3 ) )
=> ( ( divide_divide @ A @ B2 @ C3 )
= A3 ) ) ) ) ).
% divide_eq_imp
thf(fact_1353_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C3 )
= B2 )
=> ( A3
= ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_1354_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C3 )
= A3 )
= ( B2
= ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1355_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( A3
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( times_times @ A @ A3 @ C3 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1356_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( A3 = B2 ) ) ) ) ).
% right_inverse_eq
thf(fact_1357_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% divide_numeral_1
thf(fact_1358_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_1359_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_1360_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_1361_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1362_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1363_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_1364_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_1365_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W2: A,Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less @ A @ W2 @ Z3 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_1366_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W2: A,Z3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z3 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less
thf(fact_1367_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,W2: A,Z3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_le
thf(fact_1368_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_1369_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1370_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1371_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_1372_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1373_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1374_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_1375_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ X @ ( times_times @ A @ Z3 @ Y ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ Z3 ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1376_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z3: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ ( times_times @ A @ Z3 @ Y ) @ X )
=> ( ord_less @ A @ Z3 @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1377_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1378_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1379_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_1380_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_1381_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C3 )
= ( numeral_numeral @ A @ W2 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1382_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ( numeral_numeral @ A @ W2 )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1383_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
= B2 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1384_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
= A3 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1385_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z3: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z3 ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1386_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X: A,Z3: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ Z3 )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z3 @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_1387_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z3: A,X: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z3 @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_1388_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1389_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z3 ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y @ Z3 ) ) @ Z3 ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1390_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
= A3 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A3 @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1391_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z3: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z3 ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1392_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1393_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z3 ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y @ Z3 ) ) @ Z3 ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1394_card__Un__le,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B6 ) ) ) ).
% card_Un_le
thf(fact_1395_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_1396_nat__mult__div__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( divide_divide @ nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1397_Icc__eq__insert__lb__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or1337092689740270186AtMost @ nat @ M2 @ N )
= ( insert @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_1398_atLeastAtMostSuc__conv,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) )
= ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_1399_atLeastAtMost__insertL,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( insert @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_1400_subset__eq__atLeast0__atMost__finite,axiom,
! [N7: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1401_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_1402_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_1403_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_1404_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_1405_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_1406_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_1407_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_1408_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z3 @ Y ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ Z3 ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1409_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z3: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ Y ) @ X )
=> ( ord_less_eq @ A @ Z3 @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1410_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_1411_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_1412_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_1413_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1414_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1415_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z3: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z3 ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z3 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1416_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z3: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z3 ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z3 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1417_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_1418_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_1419_four__x__squared,axiom,
! [X: real] :
( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% four_x_squared
thf(fact_1420_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_1421_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_1422_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_1423_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_1424_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_1425_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V3: A,R3: A,S2: A] :
( ( ord_less_eq @ A @ U @ V3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R3 )
=> ( ( ord_less_eq @ A @ R3 @ S2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R3 @ ( minus_minus @ A @ V3 @ U ) ) @ S2 ) ) @ V3 ) ) ) ) ) ).
% scaling_mono
thf(fact_1426_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
=> ~ ! [N3: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ E3 ) ) ) ).
% nat_approx_posE
thf(fact_1427_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_1428_triangle__def,axiom,
( nat_triangle
= ( ^ [N5: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N5 @ ( suc @ N5 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_1429_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,X: A,Y: A] :
( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ X @ Y ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_1430_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_1431_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_1432_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_1433_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_1434_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_1435_insert_H__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_set_vebt @ ( vEBT_VEBT_insert @ T2 @ X ) )
= ( inf_inf @ ( set @ nat ) @ ( sup_sup @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( 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_1436_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1437_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1438_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_1439_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_1440_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_1441_zdiv__numeral__Bit0,axiom,
! [V3: num,W2: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit0 @ V3 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V3 ) @ ( numeral_numeral @ int @ W2 ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_1442_real__divide__square__eq,axiom,
! [R3: real,A3: real] :
( ( divide_divide @ real @ ( times_times @ real @ R3 @ A3 ) @ ( times_times @ real @ R3 @ R3 ) )
= ( divide_divide @ real @ A3 @ R3 ) ) ).
% real_divide_square_eq
thf(fact_1443_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1444_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_1445_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_1446_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_1447_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( divide_divide @ nat @ M2 @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_1448_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_1449_zdiv__zmult2__eq,axiom,
! [C3: int,A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C3 )
=> ( ( divide_divide @ int @ A3 @ ( times_times @ int @ B2 @ C3 ) )
= ( divide_divide @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C3 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1450_div__neg__pos__less0,axiom,
! [A3: int,B2: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_neg_pos_less0
thf(fact_1451_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1452_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1453_zdiv__int,axiom,
! [A3: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A3 @ B2 ) )
= ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zdiv_int
thf(fact_1454_atLeastAtMostPlus1__int__conv,axiom,
! [M2: int,N: int] :
( ( ord_less_eq @ int @ M2 @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
=> ( ( set_or1337092689740270186AtMost @ int @ M2 @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
= ( insert @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ ( set_or1337092689740270186AtMost @ int @ M2 @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_1455_nonneg1__imp__zdiv__pos__iff,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
= ( ( ord_less_eq @ int @ B2 @ A3 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1456_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1457_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
= ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1458_pos__imp__zdiv__pos__iff,axiom,
! [K2: int,I2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I2 @ K2 ) )
= ( ord_less_eq @ int @ K2 @ I2 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1459_div__nonpos__pos__le0,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1460_div__nonneg__neg__le0,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1461_div__positive__int,axiom,
! [L: int,K2: int] :
( ( ord_less_eq @ int @ L @ K2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K2 @ L ) ) ) ) ).
% div_positive_int
thf(fact_1462_div__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K2 @ L ) )
= ( ( K2
= ( zero_zero @ int ) )
| ( L
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
| ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% div_int_pos_iff
thf(fact_1463_zdiv__mono2__neg,axiom,
! [A3: int,B3: int,B2: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1464_zdiv__mono1__neg,axiom,
! [A3: int,A4: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ A4 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A4 @ B2 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1465_zdiv__eq__0__iff,axiom,
! [I2: int,K2: int] :
( ( ( divide_divide @ int @ I2 @ K2 )
= ( zero_zero @ int ) )
= ( ( K2
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
& ( ord_less @ int @ I2 @ K2 ) )
| ( ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) )
& ( ord_less @ int @ K2 @ I2 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1466_zdiv__mono2,axiom,
! [A3: int,B3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A3 @ B3 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1467_zdiv__mono1,axiom,
! [A3: int,A4: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ A4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A4 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_1468_int__div__less__self,axiom,
! [X: int,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K2 )
=> ( ord_less @ int @ ( divide_divide @ int @ X @ K2 ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1469_periodic__finite__ex,axiom,
! [D3: int,P2: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int,K: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K @ D3 ) ) ) )
=> ( ( ? [X7: int] : ( P2 @ X7 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ( P2 @ X4 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_1470_split__zdiv,axiom,
! [P2: int > $o,N: int,K2: int] :
( ( P2 @ ( divide_divide @ int @ N @ K2 ) )
= ( ( ( K2
= ( zero_zero @ int ) )
=> ( P2 @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ! [I: int,J: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J )
& ( ord_less @ int @ J @ K2 )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I ) @ J ) ) )
=> ( P2 @ I ) ) )
& ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ! [I: int,J: int] :
( ( ( ord_less @ int @ K2 @ J )
& ( ord_less_eq @ int @ J @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I ) @ J ) ) )
=> ( P2 @ I ) ) ) ) ) ).
% split_zdiv
thf(fact_1471_int__div__neg__eq,axiom,
! [A3: int,B2: int,Q2: int,R3: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R3 )
=> ( ( divide_divide @ int @ A3 @ B2 )
= Q2 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1472_int__div__pos__eq,axiom,
! [A3: int,B2: int,Q2: int,R3: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ R3 @ B2 )
=> ( ( divide_divide @ int @ A3 @ B2 )
= Q2 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1473_cpmi,axiom,
! [D5: int,P2: int > $o,P6: int > $o,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa ) ) ) )
=> ( ( P2 @ X3 )
=> ( P2 @ ( minus_minus @ int @ X3 @ D5 ) ) ) )
=> ( ! [X3: int,K: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K @ D5 ) ) ) )
=> ( ( ? [X7: int] : ( P2 @ X7 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ( P6 @ X4 ) )
| ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ? [Y4: int] :
( ( member @ int @ Y4 @ B6 )
& ( P2 @ ( plus_plus @ int @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_1474_cppi,axiom,
! [D5: int,P2: int > $o,P6: int > $o,A6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D5 )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A6 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa ) ) ) )
=> ( ( P2 @ X3 )
=> ( P2 @ ( plus_plus @ int @ X3 @ D5 ) ) ) )
=> ( ! [X3: int,K: int] :
( ( P6 @ X3 )
= ( P6 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K @ D5 ) ) ) )
=> ( ( ? [X7: int] : ( P2 @ X7 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ( P6 @ X4 ) )
| ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D5 ) )
& ? [Y4: int] :
( ( member @ int @ Y4 @ A6 )
& ( P2 @ ( minus_minus @ int @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_1475_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_1476_div__pos__geq,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K2 )
=> ( ( divide_divide @ int @ K2 @ L )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K2 @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_1477_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_1478_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_1479_div__mult2__eq,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( divide_divide @ nat @ M2 @ ( times_times @ nat @ N @ Q2 ) )
= ( divide_divide @ nat @ ( divide_divide @ nat @ M2 @ N ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_1480_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_1481_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_1482_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( divide_divide @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_1483_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ( ord_less @ nat @ M2 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1484_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_1485_less__mult__imp__div__less,axiom,
! [M2: nat,I2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( times_times @ nat @ I2 @ N ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ N ) @ I2 ) ) ).
% less_mult_imp_div_less
thf(fact_1486_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_1487_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_1488_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_1489_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_1490_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_1491_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_1492_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_1493_div__less__iff__less__mult,axiom,
! [Q2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q2 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ Q2 ) @ N )
= ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1494_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1495_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ( divide_divide @ nat @ M2 @ N )
= M2 )
= ( N
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_1496_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M6: nat,N5: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M6 @ N5 )
| ( N5
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M6 @ N5 ) @ N5 ) ) ) ) ) ).
% div_if
thf(fact_1497_div__nat__eqI,axiom,
! [N: nat,Q2: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q2 ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ ( suc @ Q2 ) ) )
=> ( ( divide_divide @ nat @ M2 @ N )
= Q2 ) ) ) ).
% div_nat_eqI
thf(fact_1498_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q2 )
=> ( ( ord_less_eq @ nat @ M2 @ ( divide_divide @ nat @ N @ Q2 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M2 @ Q2 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1499_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_1500_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_1501_split__div,axiom,
! [P2: nat > $o,M2: nat,N: nat] :
( ( P2 @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P2 @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I: nat,J: nat] :
( ( ord_less @ nat @ J @ N )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I ) @ J ) )
=> ( P2 @ I ) ) ) ) ) ) ).
% split_div
thf(fact_1502_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_1503_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_1504_split__div_H,axiom,
! [P2: nat > $o,M2: nat,N: nat] :
( ( P2 @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
& ( P2 @ ( zero_zero @ nat ) ) )
| ? [Q5: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q5 ) @ M2 )
& ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ ( suc @ Q5 ) ) )
& ( P2 @ Q5 ) ) ) ) ).
% split_div'
thf(fact_1505_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_1506_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_1507_linear__plus__1__le__power,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) @ ( one_one @ real ) ) @ ( power_power @ real @ ( plus_plus @ real @ X @ ( one_one @ real ) ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_1508_succ__list__to__short,axiom,
! [Deg: nat,Mi: nat,X: nat,TreeList2: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( 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 @ TreeList2 @ Summary ) @ X )
= ( none @ nat ) ) ) ) ) ).
% succ_list_to_short
thf(fact_1509_pred__list__to__short,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X @ Ma )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( 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 @ TreeList2 @ Summary ) @ X )
= ( none @ nat ) ) ) ) ) ).
% pred_list_to_short
thf(fact_1510_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_1511_case4_I11_J,axiom,
( ( mi != ma )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ( ( vEBT_VEBT_high @ ma @ na )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList2 @ I4 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ na )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ na ) ) )
=> ( ( ord_less @ nat @ mi @ X5 )
& ( ord_less_eq @ nat @ X5 @ ma ) ) ) ) ) ) ).
% case4(11)
thf(fact_1512_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_1513_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z3 )
= ( ( ord_less_eq @ A @ X @ Z3 )
& ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% le_sup_iff
thf(fact_1514_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_1515_bit__split__inv,axiom,
! [X: nat,D3: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D3 ) @ ( vEBT_VEBT_low @ X @ D3 ) @ D3 )
= X ) ).
% bit_split_inv
thf(fact_1516_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% finite_atLeastAtMost_int
thf(fact_1517_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X4: nat,N5: nat] : ( divide_divide @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ).
% high_def
thf(fact_1518_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_1519_high__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X ) @ N )
= Y ) ) ).
% high_inv
thf(fact_1520_low__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X ) @ N )
= X ) ) ).
% low_inv
thf(fact_1521_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z3 ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% le_inf_iff
thf(fact_1522_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_1523_both__member__options__ding,axiom,
! [Info2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% both_member_options_ding
thf(fact_1524_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
| ( X = Mi )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_1525_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X = Mi )
| ( X = Ma )
| ( ( ord_less @ nat @ X @ Ma )
& ( ord_less @ nat @ Mi @ X )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_1526_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_1527_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N: nat,M2: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M2 ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_1528_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N: nat,M2: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M2 ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_1529_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_sup_ord(2)
thf(fact_1530_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_sup_ord(1)
thf(fact_1531_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_le1
thf(fact_1532_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_le2
thf(fact_1533_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A3 )
=> ~ ( ord_less_eq @ A @ X @ B2 ) ) ) ) ).
% le_infE
thf(fact_1534_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_1535_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_1536_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% le_infI1
thf(fact_1537_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% le_infI2
thf(fact_1538_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_1539_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_1540_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F3: A > A > A,X: A,Y: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( F3 @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( F3 @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Z )
=> ( ord_less_eq @ A @ X3 @ ( F3 @ Y3 @ Z ) ) ) )
=> ( ( inf_inf @ A @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_1541_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y4: A] :
( ( inf_inf @ A @ X4 @ Y4 )
= X4 ) ) ) ) ).
% le_iff_inf
thf(fact_1542_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_1543_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_1544_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( inf_inf @ A @ X @ Y )
= X ) ) ) ).
% inf_absorb1
thf(fact_1545_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( inf_inf @ A @ X @ Y )
= Y ) ) ) ).
% inf_absorb2
thf(fact_1546_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_1547_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_1548_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Z3 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z3 ) ) ) ) ) ).
% inf_greatest
thf(fact_1549_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B5: A] :
( A7
= ( inf_inf @ A @ A7 @ B5 ) ) ) ) ) ).
% inf.order_iff
thf(fact_1550_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_1551_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_1552_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B5: A] :
( ( inf_inf @ A @ A7 @ B5 )
= A7 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_1553_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A7: A] :
( ( inf_inf @ A @ A7 @ B5 )
= B5 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_1554_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_1555_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_1556_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_1557_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_1558_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B5: A] :
( ( sup_sup @ A @ A7 @ B5 )
= B5 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_1559_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A7: A] :
( ( sup_sup @ A @ A7 @ B5 )
= A7 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_1560_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_1561_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_1562_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A7: A] :
( A7
= ( sup_sup @ A @ A7 @ B5 ) ) ) ) ) ).
% sup.order_iff
thf(fact_1563_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_1564_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_1565_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_1566_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( sup_sup @ A @ X @ Y )
= X ) ) ) ).
% sup_absorb1
thf(fact_1567_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_1568_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_1569_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F3: A > A > A,X: A,Y: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ ( F3 @ Y3 @ Z ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_1570_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_1571_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_1572_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y4: A] :
( ( sup_sup @ A @ X4 @ Y4 )
= Y4 ) ) ) ) ).
% le_iff_sup
thf(fact_1573_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A,Z3: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z3 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z3 ) @ X ) ) ) ) ).
% sup_least
thf(fact_1574_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_1575_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_1576_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_1577_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_1578_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge2
thf(fact_1579_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge1
thf(fact_1580_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X ) ) ) ) ).
% le_supI
thf(fact_1581_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A3 @ X )
=> ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% le_supE
thf(fact_1582_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_1583_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_1584_invar__vebt_Ointros_I4_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2 = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ 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 @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_1585_invar__vebt_Ointros_I5_J,axiom,
! [TreeList2: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ 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 @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_1586_invar__vebt_Ocases,axiom,
! [A12: vEBT_VEBT,A23: nat] :
( ( vEBT_invar_vebt @ A12 @ A23 )
=> ( ( ? [A5: $o,B4: $o] :
( A12
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( A23
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N3 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N3 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_1587_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A13: vEBT_VEBT,A24: nat] :
( ( ? [A7: $o,B5: $o] :
( A13
= ( vEBT_Leaf @ A7 @ B5 ) )
& ( A24
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList4: list @ vEBT_VEBT,N5: nat,Summary4: vEBT_VEBT] :
( ( A13
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A24 @ TreeList4 @ Summary4 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ( vEBT_invar_vebt @ X4 @ N5 ) )
& ( vEBT_invar_vebt @ Summary4 @ N5 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList4 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) )
& ( A24
= ( plus_plus @ nat @ N5 @ N5 ) )
& ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X7 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
| ? [TreeList4: list @ vEBT_VEBT,N5: nat,Summary4: vEBT_VEBT] :
( ( A13
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A24 @ TreeList4 @ Summary4 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ( vEBT_invar_vebt @ X4 @ N5 ) )
& ( vEBT_invar_vebt @ Summary4 @ ( suc @ N5 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList4 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N5 ) ) )
& ( A24
= ( plus_plus @ nat @ N5 @ ( suc @ N5 ) ) )
& ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X7 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
| ? [TreeList4: list @ vEBT_VEBT,N5: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A13
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A24 @ TreeList4 @ Summary4 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ( vEBT_invar_vebt @ X4 @ N5 ) )
& ( vEBT_invar_vebt @ Summary4 @ N5 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList4 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) )
& ( A24
= ( plus_plus @ nat @ N5 @ N5 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary4 @ I ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A24 ) )
& ( ( Mi3 != Ma3 )
=> ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N5 )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I ) @ ( vEBT_VEBT_low @ Ma3 @ N5 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N5 )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I ) @ ( vEBT_VEBT_low @ X4 @ N5 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList4: list @ vEBT_VEBT,N5: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A13
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A24 @ TreeList4 @ Summary4 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ( vEBT_invar_vebt @ X4 @ N5 ) )
& ( vEBT_invar_vebt @ Summary4 @ ( suc @ N5 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList4 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N5 ) ) )
& ( A24
= ( plus_plus @ nat @ N5 @ ( suc @ N5 ) ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N5 ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary4 @ I ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList4 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A24 ) )
& ( ( Mi3 != Ma3 )
=> ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N5 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N5 )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I ) @ ( vEBT_VEBT_low @ Ma3 @ N5 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N5 )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ I ) @ ( vEBT_VEBT_low @ X4 @ N5 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_1588_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z3 ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) ) ) ) ).
% distrib_inf_le
thf(fact_1589_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z3 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% distrib_sup_le
thf(fact_1590_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N5: nat,TreeList4: list @ vEBT_VEBT,X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ X4 @ N5 ) ) @ ( vEBT_VEBT_low @ X4 @ N5 ) ) ) ) ).
% in_children_def
thf(fact_1591_nested__mint,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va3: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( N
= ( suc @ ( suc @ Va3 ) ) )
=> ( ~ ( 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 @ Va3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide @ nat @ Va3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ).
% nested_mint
thf(fact_1592_summaxma,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
=> ( ( Mi != Ma )
=> ( ( the2 @ nat @ ( vEBT_vebt_maxt @ Summary ) )
= ( vEBT_VEBT_high @ Ma @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% summaxma
thf(fact_1593_set__encode__insert,axiom,
! [A6: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A6 )
=> ( ~ ( member @ nat @ N @ A6 )
=> ( ( nat_set_encode @ ( insert @ nat @ N @ A6 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A6 ) ) ) ) ) ).
% set_encode_insert
thf(fact_1594_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_1595_del__x__mi__lets__in__not__minNull,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ 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 @ TreeList2 @ ( 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 ) @ TreeList2 ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList2 @ H @ Newnode ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( 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_1596_del__x__not__mi__newnode__not__nil,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList2 @ H @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X = 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_1597_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q2: int,R3: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q2 @ R3 ) )
=> ( 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 @ Q2 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_1598_length__list__update,axiom,
! [A: $tType,Xs2: list @ A,I2: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs2 @ I2 @ X ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_list_update
thf(fact_1599_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_1600_list__update__beyond,axiom,
! [A: $tType,Xs2: list @ A,I2: nat,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ I2 )
=> ( ( list_update @ A @ Xs2 @ I2 @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_1601_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_1602_nth__list__update__eq,axiom,
! [A: $tType,I2: nat,Xs2: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_1603_set__swap,axiom,
! [A: $tType,I2: nat,Xs2: list @ A,J2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs2 @ I2 @ ( nth @ A @ Xs2 @ J2 ) ) @ J2 @ ( nth @ A @ Xs2 @ I2 ) ) )
= ( set2 @ A @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_1604_unique__quotient,axiom,
! [A3: int,B2: int,Q2: int,R3: int,Q4: int,R5: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R3 ) )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R5 ) )
=> ( Q2 = Q4 ) ) ) ).
% unique_quotient
thf(fact_1605_unique__remainder,axiom,
! [A3: int,B2: int,Q2: int,R3: int,Q4: int,R5: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R3 ) )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R5 ) )
=> ( R3 = R5 ) ) ) ).
% unique_remainder
thf(fact_1606_zip__update,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,I2: nat,X: A,Ys: list @ B,Y: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs2 @ I2 @ X ) @ ( list_update @ B @ Ys @ I2 @ Y ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) @ I2 @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_update
thf(fact_1607_option_Osel,axiom,
! [A: $tType,X2: A] :
( ( the2 @ A @ ( some @ A @ X2 ) )
= X2 ) ).
% option.sel
thf(fact_1608_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_1609_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_1610_div__int__unique,axiom,
! [K2: int,L: int,Q2: int,R3: int] :
( ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q2 @ R3 ) )
=> ( ( divide_divide @ int @ K2 @ L )
= Q2 ) ) ).
% div_int_unique
thf(fact_1611_list__update__code_I3_J,axiom,
! [A: $tType,X: A,Xs2: list @ A,I2: nat,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs2 ) @ ( suc @ I2 ) @ Y )
= ( cons @ A @ X @ ( list_update @ A @ Xs2 @ I2 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1612_list__update__code_I2_J,axiom,
! [A: $tType,X: A,Xs2: list @ A,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs2 ) @ ( zero_zero @ nat ) @ Y )
= ( cons @ A @ Y @ Xs2 ) ) ).
% list_update_code(2)
thf(fact_1613_set__encode__eq,axiom,
! [A6: set @ nat,B6: set @ nat] :
( ( finite_finite2 @ nat @ A6 )
=> ( ( finite_finite2 @ nat @ B6 )
=> ( ( ( nat_set_encode @ A6 )
= ( nat_set_encode @ B6 ) )
= ( A6 = B6 ) ) ) ) ).
% set_encode_eq
thf(fact_1614_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_1615_eucl__rel__int__dividesI,axiom,
! [L: int,K2: int,Q2: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K2
= ( times_times @ int @ Q2 @ L ) )
=> ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q2 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_1616_set__update__memI,axiom,
! [A: $tType,N: nat,Xs2: list @ A,X: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1617_list__update__same__conv,axiom,
! [A: $tType,I2: nat,Xs2: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( list_update @ A @ Xs2 @ I2 @ X )
= Xs2 )
= ( ( nth @ A @ Xs2 @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_1618_nth__list__update,axiom,
! [A: $tType,I2: nat,Xs2: list @ A,J2: nat,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( I2 = J2 )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I2 @ X ) @ J2 )
= X ) )
& ( ( I2 != J2 )
=> ( ( nth @ A @ ( list_update @ A @ Xs2 @ I2 @ X ) @ J2 )
= ( nth @ A @ Xs2 @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_1619_set__encode__inf,axiom,
! [A6: set @ nat] :
( ~ ( finite_finite2 @ nat @ A6 )
=> ( ( nat_set_encode @ A6 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_1620_eucl__rel__int__iff,axiom,
! [K2: int,L: int,Q2: int,R3: int] :
( ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q2 @ R3 ) )
= ( ( K2
= ( plus_plus @ int @ ( times_times @ int @ L @ Q2 ) @ R3 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
& ( ord_less @ int @ R3 @ L ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ R3 )
& ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( Q2
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_1621_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q2: int,R3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R3 ) )
=> ( 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 @ Q2 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R3 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_1622_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( ord_less @ nat @ X @ Mi )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ ( ord_max @ nat @ Mi @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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_1623_insert__simp__norm,axiom,
! [X: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( ord_less @ nat @ Mi @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( ord_max @ nat @ X @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_1624_pred__less__length__list,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X @ Ma )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X ) @ ( 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 @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% pred_less_length_list
thf(fact_1625_pred__lesseq__max,axiom,
! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X @ Ma )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X ) @ ( 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 @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% pred_lesseq_max
thf(fact_1626_succ__greatereq__min,axiom,
! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% succ_greatereq_min
thf(fact_1627_succ__less__length__list,axiom,
! [Deg: nat,Mi: nat,X: nat,TreeList2: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% succ_less_length_list
thf(fact_1628_del__in__range,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_eq @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ 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 @ TreeList2 @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( 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 @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( if @ nat
@ ( ( ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ 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 @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( if @ nat
@ ( ( ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).
% del_in_range
thf(fact_1629_del__x__mi,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,L: nat] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ 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 @ TreeList2 @ ( 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 ) @ TreeList2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ 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 @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ 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 @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ H ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% del_x_mi
thf(fact_1630_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_1631_pred__empty,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y4: nat] :
( ( vEBT_vebt_member @ T2 @ Y4 )
& ( ord_less @ nat @ Y4 @ X ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% pred_empty
thf(fact_1632_succ__empty,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_succ @ T2 @ X )
= ( none @ nat ) )
= ( ( collect @ nat
@ ^ [Y4: nat] :
( ( vEBT_vebt_member @ T2 @ Y4 )
& ( ord_less @ nat @ X @ Y4 ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% succ_empty
thf(fact_1633_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_1634_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_1635_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_1636_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% max_bot
thf(fact_1637_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% max_bot2
thf(fact_1638_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_1639_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_1640_max__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% max_nat.left_neutral
thf(fact_1641_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_1642_max__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% max_nat.right_neutral
thf(fact_1643_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_1644_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_1645_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N5: nat] : ( ord_less_eq @ nat @ N5 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1646_max__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ A @ V3 ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(1)
thf(fact_1647_max__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(3)
thf(fact_1648_max__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(4)
thf(fact_1649_max__0__1_I2_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_max @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% max_0_1(2)
thf(fact_1650_max__0__1_I1_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_max @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% max_0_1(1)
thf(fact_1651_max__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(5)
thf(fact_1652_max__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(6)
thf(fact_1653_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I: nat] : ( ord_less_eq @ nat @ I @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1654_del__x__not__mia,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X = 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 @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
@ ( vEBT_vebt_delete @ Summary @ H ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X = 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 @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ H ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% del_x_not_mia
thf(fact_1655_del__x__not__mi__new__node__nil,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList2 @ H @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X = 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_1656_del__x__not__mi,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,Newlist: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Mi @ X )
& ( ord_less_eq @ nat @ X @ Ma ) )
=> ( ( Mi != Ma )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= H )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList2 @ H @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Mi
@ ( if @ nat @ ( X = 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 @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X = 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_1657_del__x__mia,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ 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 @ TreeList2 @ Summary ) @ X )
= ( 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( 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 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ Summary ) ) ) ) ) ) ).
% del_x_mia
thf(fact_1658_del__x__mi__lets__in__minNull,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT,Sn: vEBT_VEBT] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ 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 @ TreeList2 @ ( 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 ) @ TreeList2 ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList2 @ 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 @ TreeList2 @ Summary ) @ X )
= ( 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_1659_del__x__mi__lets__in,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list @ vEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list @ vEBT_VEBT] :
( ( ( X = Mi )
& ( ord_less @ nat @ X @ 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 @ TreeList2 @ ( 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 ) @ TreeList2 ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ H ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList2 @ H @ Newnode ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( 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 @ TreeList2 @ Summary ) @ X )
= ( 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_1660_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X @ Y ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_max
thf(fact_1661_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A7: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A7 @ B5 ) @ B5 @ A7 ) ) ) ) ).
% max_def_raw
thf(fact_1662_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_1663_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H2: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_1664_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X4: A] : X4 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_1665_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P2 @ K3 )
& ( ord_less @ nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1666_finite__less__ub,axiom,
! [F3: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F3 @ N3 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N5: nat] : ( ord_less_eq @ nat @ ( F3 @ N5 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1667_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_1668_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_1669_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_1670_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_1671_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_1672_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A7: A] :
( A7
= ( ord_max @ A @ A7 @ B5 ) ) ) ) ) ).
% max.order_iff
thf(fact_1673_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_1674_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_1675_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less_eq @ A @ Z3 @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less_eq @ A @ Z3 @ X )
| ( ord_less_eq @ A @ Z3 @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_1676_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A7: A] :
( ( ord_max @ A @ A7 @ B5 )
= A7 ) ) ) ) ).
% max.absorb_iff1
thf(fact_1677_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B5: A] :
( ( ord_max @ A @ A7 @ B5 )
= B5 ) ) ) ) ).
% max.absorb_iff2
thf(fact_1678_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_1679_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_1680_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_max @ A @ X @ Y )
= Y ) ) ) ).
% max_absorb2
thf(fact_1681_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_max @ A @ X @ Y )
= X ) ) ) ).
% max_absorb1
thf(fact_1682_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A7: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A7 @ B5 ) @ B5 @ A7 ) ) ) ) ).
% max_def
thf(fact_1683_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y ) @ Z3 )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( plus_plus @ A @ Y @ Z3 ) ) ) ) ).
% max_add_distrib_left
thf(fact_1684_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y @ Z3 ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ).
% max_add_distrib_right
thf(fact_1685_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X @ Y ) @ Z3 )
= ( ord_max @ A @ ( minus_minus @ A @ X @ Z3 ) @ ( minus_minus @ A @ Y @ Z3 ) ) ) ) ).
% max_diff_distrib_left
thf(fact_1686_nat__add__max__left,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M2 @ N ) @ Q2 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ Q2 ) @ ( plus_plus @ nat @ N @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_1687_nat__add__max__right,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( plus_plus @ nat @ M2 @ ( ord_max @ nat @ N @ Q2 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( plus_plus @ nat @ M2 @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_1688_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_1689_nat__mult__max__left,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M2 @ N ) @ Q2 )
= ( ord_max @ nat @ ( times_times @ nat @ M2 @ Q2 ) @ ( times_times @ nat @ N @ Q2 ) ) ) ).
% nat_mult_max_left
thf(fact_1690_nat__mult__max__right,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ M2 @ ( ord_max @ nat @ N @ Q2 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M2 @ N ) @ ( times_times @ nat @ M2 @ Q2 ) ) ) ).
% nat_mult_max_right
thf(fact_1691_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_1692_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A7: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1693_power__numeral__even,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z3: A,W2: num] :
( ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ ( bit0 @ W2 ) ) )
= ( times_times @ A @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) ) ) ) ).
% power_numeral_even
thf(fact_1694_card__less,axiom,
! [M5: set @ nat,I2: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M5 )
& ( ord_less @ nat @ K3 @ ( suc @ I2 ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_1695_card__less__Suc,axiom,
! [M5: set @ nat,I2: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M5 )
& ( ord_less @ nat @ K3 @ I2 ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M5 )
& ( ord_less @ nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1696_card__less__Suc2,axiom,
! [M5: set @ nat,I2: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M5 )
& ( ord_less @ nat @ K3 @ I2 ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M5 )
& ( ord_less @ nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1697_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_1698_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_1699_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_1700_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_1701_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_1702_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S2 ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_1703_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd2 ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_1704_vebt__insert_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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 ) ) ) ) ) ) )
=> ( ! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) )
=> ( Y
!= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) ) )
=> ( ! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) )
=> ( Y
!= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) ) )
=> ( ! [V2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( Y
!= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( 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 @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_1705_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
= ( ( X != Mi )
=> ( ( X != Ma )
=> ( ~ ( ord_less @ nat @ X @ Mi )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ~ ( ord_less @ nat @ Ma @ X )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_1706_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ X )
= ( ( X = Mi )
| ( X = Ma )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_1707_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList: list @ vEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList @ S ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_1708_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList: list @ vEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList @ S ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_1709_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y
= ( ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) )
=> Y )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList: list @ vEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList @ S ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_1710_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( 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,V2: nat,TreeList: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) )
=> ~ ! [V2: nat,TreeList: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_1711_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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,Va: nat,TreeList: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_1712_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Uu2: $o,Uv2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( 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,V2: nat,TreeList: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) )
=> ~ ! [V2: nat,TreeList: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_1713_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> Y )
=> ( ( ? [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) )
=> Y )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( 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,V2: nat,TreeList: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_1714_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ( ! [V2: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( ! [V2: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_1715_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y
= ( ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A5 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> Y )
=> ( ( ? [V2: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> Y )
=> ( ( ? [V2: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> Y )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( Y
= ( ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_1716_vebt__pred_Osimps_I7_J,axiom,
! [Ma: nat,X: nat,Mi: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
= ( some @ nat @ Ma ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi @ X ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_pred.simps(7)
thf(fact_1717_vebt__succ_Osimps_I6_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
= ( some @ nat @ Mi ) ) )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( 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 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_succ.simps(6)
thf(fact_1718_vebt__delete_Osimps_I7_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va3: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) ) )
& ( ~ ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( ( ( X = Mi )
& ( X = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) ) )
& ( ~ ( ( X = Mi )
& ( X = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( if @ nat
@ ( ( ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( 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 @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va3 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi ) @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
@ ( if @ nat
@ ( ( ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va3 ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = 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 @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) ) ) ) ) ) ) ).
% vebt_delete.simps(7)
thf(fact_1719_vebt__delete_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X @ Xa2 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ B4 ) ) ) )
=> ( ! [A5: $o] :
( ? [B4: $o] :
( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ A5 @ $false ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ? [N3: nat] :
( Xa2
= ( suc @ ( suc @ N3 ) ) )
=> ( Y
!= ( vEBT_Leaf @ A5 @ B4 ) ) ) )
=> ( ! [Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList @ Summary2 ) )
=> ( Y
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList @ Summary2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst: list @ vEBT_VEBT,Smry: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr: list @ vEBT_VEBT,Sm: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ~ ( ( ( ( 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 @ Va ) ) @ TreeList @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( ( ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) )
& ( ~ ( ( 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) )
= 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.elims
thf(fact_1720_vebt__succ_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option @ nat] :
( ( ( vEBT_vebt_succ @ X @ Xa2 )
= Y )
=> ( ! [Uu2: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ B4 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ~ ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ( ( ? [Uv2: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ Uv2 @ Uw ) )
=> ( ? [N3: nat] :
( Xa2
= ( suc @ N3 ) )
=> ( Y
!= ( none @ nat ) ) ) )
=> ( ( ? [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V2: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V2: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ~ ( ( ( 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 @ 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.elims
thf(fact_1721_vebt__pred_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option @ nat] :
( ( ( vEBT_vebt_pred @ X @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
!= ( none @ nat ) ) ) )
=> ( ! [A5: $o] :
( ? [Uw: $o] :
( X
= ( vEBT_Leaf @ A5 @ Uw ) )
=> ( ( Xa2
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( ? [Va: nat] :
( Xa2
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( B4
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B4
=> ( ( A5
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A5
=> ( Y
= ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uy: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V2: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vd @ Ve ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V2: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ~ ( ( ( 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 @ 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.elims
thf(fact_1722_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_1723_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_1724_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_1725_vebt__succ_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option @ nat] :
( ( ( vEBT_vebt_succ @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ 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 @ Uu2 @ B4 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv2: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ Uv2 @ Uw ) )
=> ! [N3: nat] :
( ( Xa2
= ( suc @ N3 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw ) @ ( suc @ N3 ) ) ) ) ) )
=> ( ! [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ 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 ) ) @ Ux @ Uy @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) )
=> ( ( 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 ) @ V2 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) )
=> ( ( 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 ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( ( ( ( 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 @ 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.pelims
thf(fact_1726_vebt__pred_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option @ nat] :
( ( ( vEBT_vebt_pred @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa2
= ( zero_zero @ nat ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A5: $o,Uw: $o] :
( ( X
= ( 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] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ! [Va: nat] :
( ( Xa2
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( 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 @ Va ) ) ) ) ) ) )
=> ( ! [Uy: nat,Uz2: list @ vEBT_VEBT,Va2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ 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 ) ) @ Uy @ Uz2 @ Va2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vd @ Ve ) )
=> ( ( 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 ) @ V2 ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) )
=> ( ( 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 ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( ( ( ( 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 @ 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.pelims
thf(fact_1727_vebt__delete_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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] :
( ( X
= ( 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] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ! [N3: nat] :
( ( Xa2
= ( suc @ ( suc @ N3 ) ) )
=> ( ( 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 @ N3 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst: list @ vEBT_VEBT,Smry: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) )
=> ~ ( 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 ) @ TrLst @ Smry ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr: list @ vEBT_VEBT,Sm: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) )
=> ~ ( 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 ) ) @ Tr @ Sm ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( ( ( ( ( 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 @ Va ) ) @ TreeList @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa2 @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa2 ) )
=> ( ( ( ( Xa2 = Mi2 )
& ( Xa2 = Ma2 ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) )
& ( ~ ( ( 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa2 != Mi2 )
=> ( Xa2 = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( 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 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa2 = Mi2 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) )
= 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa2 = Mi2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) ) ) ) ) )
=> ~ ( 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 @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.pelims
thf(fact_1728_vebt__insert_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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 ) ) ) )
=> ( ! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) )
=> ( ( Y
= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) @ Xa2 ) ) ) )
=> ( ! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) )
=> ( ( Y
= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) @ Xa2 ) ) ) )
=> ( ! [V2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( ( Y
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( 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 @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( 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 @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_1729_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X4: nat,N5: nat] : ( modulo_modulo @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ).
% low_def
thf(fact_1730_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_1731_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% mod_by_0
thf(fact_1732_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_1733_bits__mod__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_0
thf(fact_1734_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_1735_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A3 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_1736_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_1737_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_1738_mod__div__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_div_trivial
thf(fact_1739_bits__mod__div__trivial,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_div_trivial
thf(fact_1740_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_1741_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_1742_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_1743_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_1744_mod__by__Suc__0,axiom,
! [M2: nat] :
( ( modulo_modulo @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_1745_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_1746_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_1747_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_1748_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_1749_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_1750_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_1751_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_1752_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_1753_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_1754_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_1755_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_1756_mod__mult__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ ( modulo_modulo @ A @ B2 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% mod_mult_eq
thf(fact_1757_mod__mult__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 @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C3 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_1758_mod__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% mod_mult_mult2
thf(fact_1759_mult__mod__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( times_times @ A @ C3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( modulo_modulo @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ).
% mult_mod_right
thf(fact_1760_mod__mult__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ B2 ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% mod_mult_left_eq
thf(fact_1761_mod__mult__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% mod_mult_right_eq
thf(fact_1762_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_1763_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_1764_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_1765_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_1766_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1767_mod__eq__self__iff__div__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ B2 )
= A3 )
= ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1768_cong__exp__iff__simps_I9_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q2: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ Q2 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_1769_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_1770_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_1771_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_1772_mod__induct,axiom,
! [P2: nat > $o,N: nat,P: nat,M2: nat] :
( ( P2 @ N )
=> ( ( ord_less @ nat @ N @ P )
=> ( ( ord_less @ nat @ M2 @ P )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ P )
=> ( ( P2 @ N3 )
=> ( P2 @ ( modulo_modulo @ nat @ ( suc @ N3 ) @ P ) ) ) )
=> ( P2 @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_1773_mod__less__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( modulo_modulo @ nat @ M2 @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_1774_gcd__nat__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [M: nat] : ( P2 @ M @ ( zero_zero @ nat ) )
=> ( ! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P2 @ N3 @ ( modulo_modulo @ nat @ M @ N3 ) )
=> ( P2 @ M @ N3 ) ) )
=> ( P2 @ M2 @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1775_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_1776_mod__eq__0D,axiom,
! [M2: nat,D3: nat] :
( ( ( modulo_modulo @ nat @ M2 @ D3 )
= ( zero_zero @ nat ) )
=> ? [Q3: nat] :
( M2
= ( times_times @ nat @ D3 @ Q3 ) ) ) ).
% mod_eq_0D
thf(fact_1777_mod__geq,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less @ nat @ M2 @ N )
=> ( ( modulo_modulo @ nat @ M2 @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ).
% mod_geq
thf(fact_1778_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_1779_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X @ ( times_times @ nat @ N @ Q1 ) )
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_1780_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_1781_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_1782_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q2: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_1783_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_1784_cong__exp__iff__simps_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q2: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_1785_cong__exp__iff__simps_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q2: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_1786_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_1787_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_1788_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_1789_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_1790_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_1791_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_1792_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_1793_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_1794_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_1795_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_1796_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_1797_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_1798_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_1799_div__less__mono,axiom,
! [A6: nat,B6: nat,N: nat] :
( ( ord_less @ nat @ A6 @ B6 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A6 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B6 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A6 @ N ) @ ( divide_divide @ nat @ B6 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1800_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
=> ( ( ord_less_eq @ nat @ Y @ X )
=> ? [Q3: nat] :
( X
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q3 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_1801_mod__eq__nat2E,axiom,
! [M2: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M2 @ Q2 )
= ( modulo_modulo @ nat @ N @ Q2 ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ~ ! [S: nat] :
( N
!= ( plus_plus @ nat @ M2 @ ( times_times @ nat @ Q2 @ S ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1802_mod__eq__nat1E,axiom,
! [M2: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M2 @ Q2 )
= ( modulo_modulo @ nat @ N @ Q2 ) )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ~ ! [S: nat] :
( M2
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q2 @ S ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1803_mod__mult2__eq,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( modulo_modulo @ nat @ M2 @ ( times_times @ nat @ N @ Q2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M2 @ N ) @ Q2 ) ) @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ).
% mod_mult2_eq
thf(fact_1804_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_1805_modulo__nat__def,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M6: nat,N5: nat] : ( minus_minus @ nat @ M6 @ ( times_times @ nat @ ( divide_divide @ nat @ M6 @ N5 ) @ N5 ) ) ) ) ).
% modulo_nat_def
thf(fact_1806_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_1807_split__mod,axiom,
! [P2: nat > $o,M2: nat,N: nat] :
( ( P2 @ ( modulo_modulo @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P2 @ M2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I: nat,J: nat] :
( ( ord_less @ nat @ J @ N )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I ) @ J ) )
=> ( P2 @ J ) ) ) ) ) ) ).
% split_mod
thf(fact_1808_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_1809_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_1810_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_1811_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_1812_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_1813_verit__le__mono__div,axiom,
! [A6: nat,B6: nat,N: nat] :
( ( ord_less @ nat @ A6 @ B6 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A6 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B6 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B6 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1814_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_1815_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_1816_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( modulo_modulo @ A @ X @ M2 ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M2 ) @ M2 ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_1817_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_1818_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_1819_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_1820_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_1821_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_1822_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_1823_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_1824_VEBT__internal_Oinsert_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_insert @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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 ) ) ) )
=> ~ ! [Info: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa2 )
=> ( Y
= ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) ) )
& ( ~ ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa2 )
=> ( Y
= ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.insert'.pelims
thf(fact_1825_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( 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 ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( ( Y
= ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) )
=> ~ ( 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 @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_1826_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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 ) ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) @ Xa2 ) ) )
=> ( ! [V2: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ Xa2 ) ) )
=> ( ! [V2: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( 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 ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_1827_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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 ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) @ Xa2 ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList: list @ vEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList @ S ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList @ S ) @ Xa2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_1828_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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 ) ) ) ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList: list @ vEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList @ S ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList @ S ) @ Xa2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_1829_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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 ) ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) @ Xa2 ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList: list @ vEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList @ S ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList @ S ) @ Xa2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_1830_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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,Va: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_1831_zmod__numeral__Bit0,axiom,
! [V3: num,W2: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit0 @ V3 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V3 ) @ ( numeral_numeral @ int @ W2 ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_1832_zmod__le__nonneg__dividend,axiom,
! [M2: int,K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M2 )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ M2 @ K2 ) @ M2 ) ) ).
% zmod_le_nonneg_dividend
thf(fact_1833_zmod__eq__0__iff,axiom,
! [M2: int,D3: int] :
( ( ( modulo_modulo @ int @ M2 @ D3 )
= ( zero_zero @ int ) )
= ( ? [Q5: int] :
( M2
= ( times_times @ int @ D3 @ Q5 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_1834_zmod__eq__0D,axiom,
! [M2: int,D3: int] :
( ( ( modulo_modulo @ int @ M2 @ D3 )
= ( zero_zero @ int ) )
=> ? [Q3: int] :
( M2
= ( times_times @ int @ D3 @ Q3 ) ) ) ).
% zmod_eq_0D
thf(fact_1835_zmod__int,axiom,
! [A3: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ A3 @ B2 ) )
= ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zmod_int
thf(fact_1836_mod__int__unique,axiom,
! [K2: int,L: int,Q2: int,R3: int] :
( ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q2 @ R3 ) )
=> ( ( modulo_modulo @ int @ K2 @ L )
= R3 ) ) ).
% mod_int_unique
thf(fact_1837_neg__mod__conj,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
& ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% neg_mod_conj
thf(fact_1838_pos__mod__conj,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A3 @ B2 ) )
& ( ord_less @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ B2 ) ) ) ).
% pos_mod_conj
thf(fact_1839_zmod__trivial__iff,axiom,
! [I2: int,K2: int] :
( ( ( modulo_modulo @ int @ I2 @ K2 )
= I2 )
= ( ( K2
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
& ( ord_less @ int @ I2 @ K2 ) )
| ( ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) )
& ( ord_less @ int @ K2 @ I2 ) ) ) ) ).
% zmod_trivial_iff
thf(fact_1840_div__mod__decomp__int,axiom,
! [A6: int,N: int] :
( A6
= ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A6 @ N ) @ N ) @ ( modulo_modulo @ int @ A6 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_1841_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_1842_mod__pos__geq,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K2 )
=> ( ( modulo_modulo @ int @ K2 @ L )
= ( modulo_modulo @ int @ ( minus_minus @ int @ K2 @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_1843_int__mod__pos__eq,axiom,
! [A3: int,B2: int,Q2: int,R3: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ R3 @ B2 )
=> ( ( modulo_modulo @ int @ A3 @ B2 )
= R3 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_1844_int__mod__neg__eq,axiom,
! [A3: int,B2: int,Q2: int,R3: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R3 )
=> ( ( modulo_modulo @ int @ A3 @ B2 )
= R3 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_1845_split__zmod,axiom,
! [P2: int > $o,N: int,K2: int] :
( ( P2 @ ( modulo_modulo @ int @ N @ K2 ) )
= ( ( ( K2
= ( zero_zero @ int ) )
=> ( P2 @ N ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ! [I: int,J: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J )
& ( ord_less @ int @ J @ K2 )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I ) @ J ) ) )
=> ( P2 @ J ) ) )
& ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ! [I: int,J: int] :
( ( ( ord_less @ int @ K2 @ J )
& ( ord_less_eq @ int @ J @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I ) @ J ) ) )
=> ( P2 @ J ) ) ) ) ) ).
% split_zmod
thf(fact_1846_zmod__zmult2__eq,axiom,
! [C3: int,A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C3 )
=> ( ( modulo_modulo @ int @ A3 @ ( times_times @ int @ B2 @ C3 ) )
= ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C3 ) ) @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_1847_split__pos__lemma,axiom,
! [K2: int,P2: int > int > $o,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( P2 @ ( divide_divide @ int @ N @ K2 ) @ ( modulo_modulo @ int @ N @ K2 ) )
= ( ! [I: int,J: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J )
& ( ord_less @ int @ J @ K2 )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I ) @ J ) ) )
=> ( P2 @ I @ J ) ) ) ) ) ).
% split_pos_lemma
thf(fact_1848_split__neg__lemma,axiom,
! [K2: int,P2: int > int > $o,N: int] :
( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ( ( P2 @ ( divide_divide @ int @ N @ K2 ) @ ( modulo_modulo @ int @ N @ K2 ) )
= ( ! [I: int,J: int] :
( ( ( ord_less @ int @ K2 @ J )
& ( ord_less_eq @ int @ J @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I ) @ J ) ) )
=> ( P2 @ I @ J ) ) ) ) ) ).
% split_neg_lemma
thf(fact_1849_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_1850_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_1851_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
=> ( ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( 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,V2: nat,TreeList: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
=> ( ( Y
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) )
=> ~ ( 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 @ V2 ) @ TreeList @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [V2: nat,TreeList: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_1852_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
=> ( ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Xa2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( 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,V2: nat,TreeList: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ 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 @ V2 ) @ TreeList @ Vc2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd ) @ Xa2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_1853_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( 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,V2: nat,TreeList: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ 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 @ V2 ) @ TreeList @ Vc2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd ) @ Xa2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( 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 @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_1854_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_1855_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H: A,Z3: A,K5: real,N: nat] :
( ( H
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z3 @ 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 @ Z3 @ H ) @ N ) @ ( power_power @ A @ Z3 @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z3 @ ( 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_1856_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_1857_artanh__0,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% artanh_0
thf(fact_1858_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X8: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N5 ) ) @ K6 ) ) )
= ( ? [N6: nat] :
! [N5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N5 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_1859_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X8: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N5 ) ) @ K6 ) ) )
= ( ? [N6: nat] :
! [N5: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N5 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_1860_norm__mult__numeral1,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [W2: num,A3: A] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ A3 ) )
= ( times_times @ real @ ( numeral_numeral @ real @ W2 ) @ ( real_V7770717601297561774m_norm @ A @ A3 ) ) ) ) ).
% norm_mult_numeral1
thf(fact_1861_norm__mult__numeral2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A,W2: num] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) )
= ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( numeral_numeral @ real @ W2 ) ) ) ) ).
% norm_mult_numeral2
thf(fact_1862_norm__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% norm_le_zero_iff
thf(fact_1863_zero__less__norm__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
= ( X
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_norm_iff
thf(fact_1864_norm__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% norm_zero
thf(fact_1865_norm__eq__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( real_V7770717601297561774m_norm @ A @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% norm_eq_zero
thf(fact_1866_norm__power__diff,axiom,
! [A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z3: A,W2: A,M2: nat] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ W2 ) @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( power_power @ A @ Z3 @ M2 ) @ ( power_power @ A @ W2 @ M2 ) ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Z3 @ W2 ) ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_1867_norm__mult,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,Y: A] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y ) )
= ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_mult
thf(fact_1868_nonzero__norm__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ) ).
% nonzero_norm_divide
thf(fact_1869_power__eq__imp__eq__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W2: A,N: nat,Z3: A] :
( ( ( power_power @ A @ W2 @ N )
= ( power_power @ A @ Z3 @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( real_V7770717601297561774m_norm @ A @ W2 )
= ( real_V7770717601297561774m_norm @ A @ Z3 ) ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_1870_norm__mult__less,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A,R3: real,Y: A,S2: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R3 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y ) ) @ ( times_times @ real @ R3 @ S2 ) ) ) ) ) ).
% norm_mult_less
thf(fact_1871_norm__mult__ineq,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_mult_ineq
thf(fact_1872_power__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W2: A,N: nat] :
( ( ( power_power @ A @ W2 @ N )
= ( one_one @ A ) )
=> ( ( ( real_V7770717601297561774m_norm @ A @ W2 )
= ( one_one @ real ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% power_eq_1_iff
thf(fact_1873_arsinh__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% arsinh_0
thf(fact_1874_vebt__buildup_Oelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
thf(fact_1875_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I5 )
& ( ( X @ I )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I5 )
& ( ( Y @ I )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I5 )
& ( ( times_times @ A @ ( X @ I ) @ ( Y @ I ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_1876_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M2: nat,X: A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_1877_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I5 )
& ( ( X @ I )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I5 )
& ( ( Y @ I )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I5 )
& ( ( plus_plus @ A @ ( X @ I ) @ ( Y @ I ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_1878_geometric__deriv__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( one_one @ real ) )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N5 ) ) @ ( power_power @ A @ Z3 @ N5 ) )
@ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_1879_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z3 ) @ ( 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 @ Z3 @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% pochhammer_double
thf(fact_1880_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ A3 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_1881_dvd__0__left__iff,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left_iff
thf(fact_1882_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_1883_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_1884_div__dvd__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ C3 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( divide_divide @ A @ C3 @ A3 ) )
= ( dvd_dvd @ A @ B2 @ C3 ) ) ) ) ) ).
% div_dvd_div
thf(fact_1885_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_1886_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu3: B] : ( zero_zero @ A )
@ A6 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_1887_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C3 @ A3 ) )
= ( dvd_dvd @ A @ B2 @ C3 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1888_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) )
= ( dvd_dvd @ A @ B2 @ C3 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1889_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1890_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1891_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_1892_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_1893_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_1894_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% dvd_div_mult_self
thf(fact_1895_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ A3 ) )
= B2 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_1896_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_1897_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_1898_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_1899_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= A3 ) ) ) ).
% unit_div_1_div_1
thf(fact_1900_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_1901_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,G3: B > A] :
( ~ ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 )
= ( zero_zero @ A ) ) ) ) ).
% sum.infinite
thf(fact_1902_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ F5 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ F5 )
= ( zero_zero @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ F5 )
=> ( ( F3 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_1903_div__diff,axiom,
! [A: $tType] :
( ( idom_modulo @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( divide_divide @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 )
= ( minus_minus @ A @ ( divide_divide @ A @ A3 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ) ).
% div_diff
thf(fact_1904_dvd__imp__mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( modulo_modulo @ A @ B2 @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% dvd_imp_mod_0
thf(fact_1905_dvd__1__left,axiom,
! [K2: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K2 ) ).
% dvd_1_left
thf(fact_1906_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_1907_nat__mult__dvd__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( ( K2
= ( zero_zero @ nat ) )
| ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1908_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_1909_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% pochhammer_Suc0
thf(fact_1910_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_1911_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_1912_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( divide_divide @ A @ B2 @ A3 ) ) ) ) ).
% unit_mult_div_div
thf(fact_1913_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% unit_div_mult_self
thf(fact_1914_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_1915_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y: A,A6: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : Y
@ A6 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ Y ) ) ) ).
% sum_constant
thf(fact_1916_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_1917_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_1918_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_1919_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A,X: A] :
( ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( A3 @ N5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N5 ) )
@ X )
= ( ( A3 @ ( zero_zero @ nat ) )
= X ) ) ) ).
% powser_sums_zero_iff
thf(fact_1920_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_1921_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_1922_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_1923_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
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ ( zero_zero @ A ) @ I ) )
@ A6 )
= ( C3 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A6 )
& ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ ( zero_zero @ A ) @ I ) )
@ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_1924_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_1925_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_1926_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_1927_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_1928_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_1929_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
@ ^ [I: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ ( zero_zero @ A ) @ I ) ) @ ( D3 @ I ) )
@ 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
@ ^ [I: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ ( zero_zero @ A ) @ I ) ) @ ( D3 @ I ) )
@ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_1930_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_1931_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_1932_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( ( numeral_numeral @ nat @ W2 )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_1933_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_1934_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_1935_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_1936_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_1937_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,G3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_1938_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,A6: set @ B] :
( ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 )
!= ( zero_zero @ A ) )
=> ~ ! [A5: B] :
( ( member @ B @ A5 @ A6 )
=> ( ( G3 @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_1939_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_1940_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R3: A,F3: B > A,A6: set @ B] :
( ( times_times @ A @ R3 @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N5: B] : ( times_times @ A @ R3 @ ( F3 @ N5 ) )
@ A6 ) ) ) ).
% sum_distrib_left
thf(fact_1941_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F3: B > A,A6: set @ B,R3: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) @ R3 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N5: B] : ( times_times @ A @ ( F3 @ N5 ) @ R3 )
@ A6 ) ) ) ).
% sum_distrib_right
thf(fact_1942_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F3: A > B,A6: set @ A,G3: C > B,B6: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A6 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G3 @ B6 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J: C] : ( times_times @ B @ ( F3 @ I ) @ ( G3 @ J ) )
@ B6 )
@ A6 ) ) ) ).
% sum_product
thf(fact_1943_dvd__antisym,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd @ nat @ M2 @ N )
=> ( ( dvd_dvd @ nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% dvd_antisym
thf(fact_1944_dvd__trans,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ C3 )
=> ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% dvd_trans
thf(fact_1945_dvd__refl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ A3 @ A3 ) ) ).
% dvd_refl
thf(fact_1946_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A7: A,B5: A] :
( ( A7
= ( zero_zero @ A ) )
=> ( B5
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_1947_dvd__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
=> ( A3
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left
thf(fact_1948_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ P @ ( times_times @ A @ A3 @ B2 ) )
=> ~ ! [X3: A,Y3: A] :
( ( P
= ( times_times @ A @ X3 @ Y3 ) )
=> ( ( dvd_dvd @ A @ X3 @ A3 )
=> ~ ( dvd_dvd @ A @ Y3 @ B2 ) ) ) ) ) ).
% dvd_productE
thf(fact_1949_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
=> ? [B10: A,C6: A] :
( ( A3
= ( times_times @ A @ B10 @ C6 ) )
& ( dvd_dvd @ A @ B10 @ B2 )
& ( dvd_dvd @ A @ C6 @ C3 ) ) ) ) ).
% division_decomp
thf(fact_1950_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ~ ! [K: A] :
( A3
!= ( times_times @ A @ B2 @ K ) ) ) ) ).
% dvdE
thf(fact_1951_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A3: A,B2: A,K2: A] :
( ( A3
= ( times_times @ A @ B2 @ K2 ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% dvdI
thf(fact_1952_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B5: A,A7: A] :
? [K3: A] :
( A7
= ( times_times @ A @ B5 @ K3 ) ) ) ) ) ).
% dvd_def
thf(fact_1953_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C3 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% dvd_mult
thf(fact_1954_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% dvd_mult2
thf(fact_1955_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
=> ( dvd_dvd @ A @ A3 @ C3 ) ) ) ).
% dvd_mult_left
thf(fact_1956_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) ) ) ).
% dvd_triv_left
thf(fact_1957_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C3 @ D3 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_1958_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
=> ( dvd_dvd @ A @ B2 @ C3 ) ) ) ).
% dvd_mult_right
thf(fact_1959_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) ) ) ).
% dvd_triv_right
thf(fact_1960_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_1961_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_1962_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_1963_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_1964_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% unit_imp_dvd
thf(fact_1965_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% one_dvd
thf(fact_1966_dvd__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( dvd_dvd @ A @ X @ Z3 )
=> ( dvd_dvd @ A @ X @ ( minus_minus @ A @ Y @ Z3 ) ) ) ) ) ).
% dvd_diff
thf(fact_1967_div__div__div__same,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [D3: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ D3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ D3 ) @ ( divide_divide @ A @ B2 @ D3 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_div_div_same
thf(fact_1968_dvd__div__eq__cancel,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ C3 )
= ( divide_divide @ A @ B2 @ C3 ) )
=> ( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( A3 = B2 ) ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1969_dvd__div__eq__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( ( divide_divide @ A @ A3 @ C3 )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( A3 = B2 ) ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1970_gcd__nat_Oextremum,axiom,
! [A3: nat] : ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_1971_gcd__nat_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
& ( ( zero_zero @ nat )
!= A3 ) ) ).
% gcd_nat.extremum_strict
thf(fact_1972_gcd__nat_Oextremum__unique,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_1973_gcd__nat_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
= ( ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) )
& ( A3
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1974_gcd__nat_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1975_dvd__mod__imp__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( dvd_dvd @ A @ C3 @ A3 ) ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_1976_dvd__mod__iff,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( dvd_dvd @ A @ C3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( dvd_dvd @ A @ C3 @ A3 ) ) ) ) ).
% dvd_mod_iff
thf(fact_1977_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A6: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) ) ) ) ).
% sum_nonneg
thf(fact_1978_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A6: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_1979_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_1980_sum__mono__inv,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F3: I6 > A,I5: set @ I6,G3: I6 > A,I2: 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 @ I2 @ I5 )
=> ( ( finite_finite2 @ I6 @ I5 )
=> ( ( F3 @ I2 )
= ( G3 @ I2 ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1981_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 )
=> ( ! [X3: nat] :
( ( member @ nat @ ( suc @ X3 ) @ A6 )
=> ( ( F3 @ ( suc @ X3 ) )
= ( G3 @ ( suc @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ A6 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ A6 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_1982_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,G3: B > A,P2: B > $o] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( P2 @ X4 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P2 @ X4 ) @ ( G3 @ X4 ) @ ( zero_zero @ A ) )
@ A6 ) ) ) ) ).
% sum.inter_filter
thf(fact_1983_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A3 ) )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_1984_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A3 ) )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
= ( ( dvd_dvd @ A @ A3 @ B2 )
& ~ ( dvd_dvd @ A @ B2 @ A3 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_1985_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S2: set @ B,T2: set @ C,G3: C > A,I2: C > B,F3: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G3 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ? [Xa: C] :
( ( member @ C @ Xa @ T2 )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ Xa ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S2 ) @ ( groups7311177749621191930dd_sum @ C @ A @ G3 @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1986_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ( ( 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_1987_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 )
=> ( ! [X3: I6] :
( ( member @ I6 @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ? [X5: I6] :
( ( member @ I6 @ X5 @ A6 )
& ( ord_less @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I6 @ A @ F3 @ A6 ) @ ( groups7311177749621191930dd_sum @ I6 @ A @ G3 @ A6 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1988_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R2: A > A > $o,S3: set @ B,H: B > A,G3: B > A] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X15: A,Y15: A,X23: A,Y23: A] :
( ( ( R2 @ X15 @ X23 )
& ( R2 @ Y15 @ Y23 ) )
=> ( R2 @ ( plus_plus @ A @ X15 @ Y15 ) @ ( plus_plus @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( R2 @ ( H @ X3 ) @ ( G3 @ X3 ) ) )
=> ( R2 @ ( groups7311177749621191930dd_sum @ B @ A @ H @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_1989_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ B,T5: set @ C,S3: set @ B,I2: C > B,J2: B > C,T6: set @ C,G3: B > A,H: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
=> ( ( I2 @ ( J2 @ A5 ) )
= A5 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
=> ( member @ C @ ( J2 @ A5 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ( J2 @ ( I2 @ B4 ) )
= B4 ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( member @ B @ ( I2 @ B4 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S5 )
=> ( ( G3 @ A5 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( H @ B4 )
= ( zero_zero @ A ) ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S3 )
=> ( ( H @ ( J2 @ A5 ) )
= ( G3 @ A5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ H @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_1990_not__is__unit__0,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% not_is_unit_0
thf(fact_1991_dvd__div__eq__0__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1992_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B2 @ A3 )
= ( times_times @ A @ C3 @ A3 ) )
= ( B2 = C3 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_1993_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A3 @ B2 )
= ( times_times @ A @ A3 @ C3 ) )
= ( B2 = C3 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_1994_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( dvd_dvd @ A @ B2 @ C3 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_1995_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_1996_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_1997_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_1998_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_1999_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C3 ) ) ) ) ).
% dvd_div_mult
thf(fact_2000_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ) ).
% div_mult_swap
thf(fact_2001_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_2002_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C3 ) @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_2003_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 )
=> ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_2004_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,D3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( dvd_dvd @ A @ D3 @ C3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( divide_divide @ A @ C3 @ D3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_2005_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= ( divide_divide @ A @ C3 @ A3 ) )
= ( B2 = C3 ) ) ) ) ).
% unit_div_cancel
thf(fact_2006_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_2007_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ C3 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_2008_mod__eq__0__iff__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% mod_eq_0_iff_dvd
thf(fact_2009_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A7: A,B5: A] :
( ( modulo_modulo @ A @ B5 @ A7 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_2010_mod__0__imp__dvd,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% mod_0_imp_dvd
thf(fact_2011_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N: nat,M2: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ M2 ) ) ) ) ) ).
% dvd_power_le
thf(fact_2012_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_2013_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_2014_dvd__minus__mod,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A3: A] : ( dvd_dvd @ A @ B2 @ ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% dvd_minus_mod
thf(fact_2015_dvd__pos__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ M2 @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 ) ) ) ).
% dvd_pos_nat
thf(fact_2016_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_2017_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% pochhammer_pos
thf(fact_2018_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_2019_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S2: set @ B,F3: B > A,I2: B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S2 )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I2 @ S2 )
=> ( ( F3 @ I2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_2020_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S2: set @ B,F3: B > A,B6: A,I2: B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S2 )
= B6 )
=> ( ( member @ B @ I2 @ S2 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ B6 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_2021_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_2022_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_2023_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_2024_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_2025_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_2026_bezout__add__nat,axiom,
! [A3: nat,B2: nat] :
? [D2: nat,X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ( ( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D2 ) )
| ( ( times_times @ nat @ B2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_nat
thf(fact_2027_bezout__lemma__nat,axiom,
! [D3: nat,A3: nat,B2: nat,X: nat,Y: nat] :
( ( dvd_dvd @ nat @ D3 @ A3 )
=> ( ( dvd_dvd @ nat @ D3 @ B2 )
=> ( ( ( ( times_times @ nat @ A3 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y ) @ D3 ) )
| ( ( times_times @ nat @ B2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y ) @ D3 ) ) )
=> ? [X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D3 @ A3 )
& ( dvd_dvd @ nat @ D3 @ ( plus_plus @ nat @ A3 @ B2 ) )
& ( ( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ Y3 ) @ D3 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y3 ) @ D3 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_2028_bezout1__nat,axiom,
! [A3: nat,B2: nat] :
? [D2: nat,X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ( ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X3 ) @ ( times_times @ nat @ B2 @ Y3 ) )
= D2 )
| ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A3 @ Y3 ) )
= D2 ) ) ) ).
% bezout1_nat
thf(fact_2029_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,G3: B > A,B6: set @ B] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( member @ B @ X4 @ B6 ) @ ( G3 @ X4 ) @ ( zero_zero @ A ) )
@ A6 ) ) ) ) ).
% sum.inter_restrict
thf(fact_2030_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3
@ ( minus_minus @ ( set @ B ) @ A6
@ ( collect @ B
@ ^ [X4: B] :
( ( G3 @ X4 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_2031_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
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_2032_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
@ ^ [I: nat] : ( G3 @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_2033_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,I2: B,F3: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( member @ B @ I2 @ I5 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
=> ( ! [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_2034_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_2035_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_2036_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_2037_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S3 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_2038_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,H: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T6 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_2039_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_2040_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T6 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_2041_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A6: set @ B,B6: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A6 ) )
=> ( ( G3 @ A5 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C5 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B6 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_2042_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A6: set @ B,B6: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A6 ) )
=> ( ( G3 @ A5 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B6 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C5 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_2043_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,H: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S3 @ T6 ) )
=> ( ( G3 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S3 @ T6 ) )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T6 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_2044_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P2: A > $o,L: A] :
( ( ? [X4: A] : ( P2 @ ( times_times @ A @ L @ X4 ) ) )
= ( ? [X4: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X4 @ ( zero_zero @ A ) ) )
& ( P2 @ X4 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_2045_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [C2: A] :
( B2
!= ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% unit_dvdE
thf(fact_2046_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A,B2: A,D3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C3 @ D3 )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= ( divide_divide @ A @ D3 @ C3 ) )
= ( ( times_times @ A @ B2 @ C3 )
= ( times_times @ A @ A3 @ D3 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_2047_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_2048_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 )
= ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_2049_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= C3 )
= ( B2
= ( times_times @ A @ C3 @ A3 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_2050_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_2051_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D5: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D5 )
=> ! [X5: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X5 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K4 @ D5 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_2052_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D5: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D5 )
=> ! [X5: A,K4: A] :
( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X5 @ T2 ) )
= ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K4 @ D5 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_2053_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= C3 )
= ( A3
= ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_2054_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( A3
= ( divide_divide @ A @ C3 @ B2 ) )
= ( ( times_times @ A @ A3 @ B2 )
= C3 ) ) ) ) ).
% unit_eq_div2
thf(fact_2055_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_2056_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% unit_div_commute
thf(fact_2057_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_2058_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_2059_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_2060_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_2061_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% pochhammer_nonneg
thf(fact_2062_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_2063_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_2064_nat__mult__dvd__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_2065_pochhammer__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_0_left
thf(fact_2066_bezout__add__strong__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [D2: nat,X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_2067_mod__greater__zero__iff__not__dvd,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M2 @ N ) )
= ( ~ ( dvd_dvd @ nat @ N @ M2 ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_2068_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M2: nat,Q2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( ( modulo_modulo @ nat @ M2 @ Q2 )
= ( modulo_modulo @ nat @ N @ Q2 ) )
= ( dvd_dvd @ nat @ Q2 @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_2069_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M2: nat,I5: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M2 @ I ) )
@ I5 )
= ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I5 ) ) ) ) ).
% sum_power_add
thf(fact_2070_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B6: set @ B,A6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B6 )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ B6 @ 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 @ B6 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_2071_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,B6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_2072_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
@ ^ [I: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_2073_finite__divisors__nat,axiom,
! [M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ M2 ) ) ) ) ).
% finite_divisors_nat
thf(fact_2074_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A] :
( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( A3 @ N5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N5 ) )
@ ( A3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_2075_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_2076_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_2077_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_2078_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_2079_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_2080_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [B4: A] :
( ( B4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B4 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= B4 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B4 )
= A3 )
=> ( ( ( times_times @ A @ A3 @ B4 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C3 @ A3 )
!= ( times_times @ A @ C3 @ B4 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_2081_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_2082_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_2083_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_2084_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M2: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ N ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_2085_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N ) ) ) ) ).
% dvd_power
thf(fact_2086_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B6: set @ A,A6: set @ A,B2: A,F3: A > B] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B6 @ A6 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F3 @ B2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X3 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A6 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ B6 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_2087_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I2: C,A6: set @ C,F3: C > B] :
( ( member @ C @ I2 @ A6 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A6 @ ( insert @ C @ I2 @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X3 ) ) )
=> ( ( finite_finite2 @ C @ A6 )
=> ( ord_less_eq @ B @ ( F3 @ I2 ) @ ( groups7311177749621191930dd_sum @ C @ B @ F3 @ A6 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_2088_div2__even__ext__nat,axiom,
! [X: nat,Y: nat] :
( ( ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Y ) )
=> ( X = Y ) ) ) ).
% div2_even_ext_nat
thf(fact_2089_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A6: set @ B,F3: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less @ A @ ( F3 @ I3 ) @ K5 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A6 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A6 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_2090_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_2091_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_2092_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_2093_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_2094_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z3 @ ( suc @ N ) )
= ( times_times @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z3 @ N ) ) ) ) ).
% pochhammer_rec'
thf(fact_2095_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_2096_dvd__minus__add,axiom,
! [Q2: nat,N: nat,R3: nat,M2: nat] :
( ( ord_less_eq @ nat @ Q2 @ N )
=> ( ( ord_less_eq @ nat @ Q2 @ ( times_times @ nat @ R3 @ M2 ) )
=> ( ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N @ Q2 ) )
= ( dvd_dvd @ nat @ M2 @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R3 @ M2 ) @ Q2 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_2097_power__dvd__imp__le,axiom,
! [I2: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I2 @ M2 ) @ ( power_power @ nat @ I2 @ N ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I2 )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_2098_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z3: A,N: nat,M2: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z3 @ ( plus_plus @ nat @ N @ M2 ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ N ) ) @ M2 ) ) ) ) ).
% pochhammer_product'
thf(fact_2099_mod__nat__eqI,axiom,
! [R3: nat,N: nat,M2: nat] :
( ( ord_less @ nat @ R3 @ N )
=> ( ( ord_less_eq @ nat @ R3 @ M2 )
=> ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M2 @ R3 ) )
=> ( ( modulo_modulo @ nat @ M2 @ N )
= R3 ) ) ) ) ).
% mod_nat_eqI
thf(fact_2100_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
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_2101_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
@ ^ [I: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I ) ) @ ( F3 @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F3 @ ( suc @ N ) ) @ ( F3 @ M2 ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_2102_sum__norm__bound,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S3: set @ B,F3: B > A,K5: real] :
( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X3 ) ) @ K5 ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S3 ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( finite_card @ B @ S3 ) ) @ K5 ) ) ) ) ).
% sum_norm_bound
thf(fact_2103_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A,P: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_2104_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_2105_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_2106_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_2107_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_2108_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_2109_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_2110_set__encode__def,axiom,
( nat_set_encode
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% set_encode_def
thf(fact_2111_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_2112_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_2113_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M2: nat,N: nat,Z3: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z3 @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ M2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_2114_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_2115_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_2116_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_2117_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_2118_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_2119_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_2120_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_2121_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_2122_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_2123_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_2124_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_2125_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
@ ^ [I: nat] : ( plus_plus @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_2126_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_2127_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_2128_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_2129_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_2130_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_2131_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_2132_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_2133_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
@ ^ [I: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I ) @ 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_2134_arith__series__nat,axiom,
! [A3: nat,D3: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I @ 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_2135_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_2136_Bernoulli__inequality__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_2137_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_2138_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,D3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I ) @ 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_2139_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_2140_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_2141_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M2: nat,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_2142_vebt__buildup_Osimps_I3_J,axiom,
! [Va3: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va3 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va3 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_2143_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_2144_power__half__series,axiom,
( sums @ real
@ ^ [N5: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N5 ) )
@ ( one_one @ real ) ) ).
% power_half_series
thf(fact_2145_sums__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( sums @ A
@ ^ [N5: nat] : ( zero_zero @ A )
@ ( zero_zero @ A ) ) ) ).
% sums_zero
thf(fact_2146_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M2: nat,Z3: A] :
( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( if @ A @ ( N5 = M2 ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z3 @ N5 ) )
@ ( power_power @ A @ Z3 @ M2 ) ) ) ).
% powser_sums_if
thf(fact_2147_sums__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A6: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ A6 )
=> ( sums @ A
@ ^ [R: nat] : ( if @ A @ ( member @ nat @ R @ A6 ) @ ( F3 @ R ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ A6 ) ) ) ) ).
% sums_If_finite_set
thf(fact_2148_sums__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P2: nat > $o,F3: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P2 ) )
=> ( sums @ A
@ ^ [R: nat] : ( if @ A @ ( P2 @ R ) @ ( F3 @ R ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( collect @ nat @ P2 ) ) ) ) ) ).
% sums_If_finite
thf(fact_2149_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N7: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N7 )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( sums @ A @ F3 @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ N7 ) ) ) ) ) ).
% sums_finite
thf(fact_2150_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N: nat,F3: nat > A,S2: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( F3 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I: nat] : ( F3 @ ( plus_plus @ nat @ I @ N ) )
@ S2 )
= ( sums @ A @ F3 @ S2 ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_2151_zdvd__mono,axiom,
! [K2: int,M2: int,T2: int] :
( ( K2
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ M2 @ T2 )
= ( dvd_dvd @ int @ ( times_times @ int @ K2 @ M2 ) @ ( times_times @ int @ K2 @ T2 ) ) ) ) ).
% zdvd_mono
thf(fact_2152_zdvd__mult__cancel,axiom,
! [K2: int,M2: int,N: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ K2 @ M2 ) @ ( times_times @ int @ K2 @ N ) )
=> ( ( K2
!= ( zero_zero @ int ) )
=> ( dvd_dvd @ int @ M2 @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_2153_zdvd__period,axiom,
! [A3: int,D3: int,X: int,T2: int,C3: int] :
( ( dvd_dvd @ int @ A3 @ D3 )
=> ( ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ X @ T2 ) )
= ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C3 @ D3 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_2154_zdvd__reduce,axiom,
! [K2: int,N: int,M2: int] :
( ( dvd_dvd @ int @ K2 @ ( plus_plus @ int @ N @ ( times_times @ int @ K2 @ M2 ) ) )
= ( dvd_dvd @ int @ K2 @ N ) ) ).
% zdvd_reduce
thf(fact_2155_sum__subtractf__nat,axiom,
! [A: $tType,A6: set @ A,G3: A > nat,F3: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ nat @ ( G3 @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ( 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_2156_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 ) ) )
& ! [Y4: A] :
( ( member @ A @ Y4 @ A6 )
=> ( ( X4 != Y4 )
=> ( ( F3 @ Y4 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_2157_sum__SucD,axiom,
! [A: $tType,F3: A > nat,A6: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A6 )
= ( suc @ N ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_2158_sum__eq__1__iff,axiom,
! [A: $tType,A6: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A6 )
= ( one_one @ nat ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ( F3 @ X4 )
= ( one_one @ nat ) )
& ! [Y4: A] :
( ( member @ A @ Y4 @ A6 )
=> ( ( X4 != Y4 )
=> ( ( F3 @ Y4 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_2159_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_2160_sum__multicount,axiom,
! [A: $tType,B: $tType,S3: set @ A,T6: set @ B,R2: A > B > $o,K2: nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T6 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I: A] :
( ( member @ A @ I @ S3 )
& ( R2 @ I @ X3 ) ) ) )
= K2 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J: B] :
( ( member @ B @ J @ T6 )
& ( R2 @ I @ J ) ) ) )
@ S3 )
= ( times_times @ nat @ K2 @ ( finite_card @ B @ T6 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_2161_mod__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K2 @ L ) )
= ( ( dvd_dvd @ int @ L @ K2 )
| ( ( L
= ( zero_zero @ int ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_2162_sum__count__set,axiom,
! [A: $tType,Xs2: list @ A,X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ X8 )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs2 ) @ X8 )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% sum_count_set
thf(fact_2163_sums__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,G3: nat > A,S2: A,T2: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G3 @ N3 ) )
=> ( ( sums @ A @ F3 @ S2 )
=> ( ( sums @ A @ G3 @ T2 )
=> ( ord_less_eq @ A @ S2 @ T2 ) ) ) ) ) ).
% sums_le
thf(fact_2164_sums__0,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A] :
( ! [N3: nat] :
( ( F3 @ N3 )
= ( zero_zero @ A ) )
=> ( sums @ A @ F3 @ ( zero_zero @ A ) ) ) ) ).
% sums_0
thf(fact_2165_sums__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I2: nat,F3: nat > A] :
( sums @ A
@ ^ [R: nat] : ( if @ A @ ( R = I2 ) @ ( F3 @ R ) @ ( zero_zero @ A ) )
@ ( F3 @ I2 ) ) ) ).
% sums_single
thf(fact_2166_sums__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,A3: A,C3: A] :
( ( sums @ A @ F3 @ A3 )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ C3 )
@ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% sums_mult2
thf(fact_2167_sums__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,A3: A,C3: A] :
( ( sums @ A @ F3 @ A3 )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ C3 @ ( F3 @ N5 ) )
@ ( times_times @ A @ C3 @ A3 ) ) ) ) ).
% sums_mult
thf(fact_2168_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_2169_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C3: A,F3: nat > A,D3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ C3 @ ( F3 @ N5 ) )
@ ( times_times @ A @ C3 @ D3 ) )
= ( sums @ A @ F3 @ D3 ) ) ) ) ).
% sums_mult_iff
thf(fact_2170_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C3: A,F3: nat > A,D3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ C3 )
@ ( times_times @ A @ D3 @ C3 ) )
= ( sums @ A @ F3 @ D3 ) ) ) ) ).
% sums_mult2_iff
thf(fact_2171_sums__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A,A3: A] :
( ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ C3 @ ( F3 @ N5 ) )
@ A3 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( sums @ A @ F3 @ ( divide_divide @ A @ A3 @ C3 ) ) ) ) ) ).
% sums_mult_D
thf(fact_2172_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S2: A] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N5: nat] : ( F3 @ ( suc @ N5 ) )
@ S2 )
=> ( sums @ A @ F3 @ S2 ) ) ) ) ).
% sums_Suc_imp
thf(fact_2173_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S2: A] :
( ( sums @ A
@ ^ [N5: nat] : ( F3 @ ( suc @ N5 ) )
@ S2 )
= ( sums @ A @ F3 @ ( plus_plus @ A @ S2 @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_2174_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,L: A] :
( ( sums @ A
@ ^ [N5: nat] : ( F3 @ ( suc @ N5 ) )
@ L )
=> ( sums @ A @ F3 @ ( plus_plus @ A @ L @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_2175_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z3: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( plus_plus @ A @ Z3 @ ( 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_2176_lemma__termdiff2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [H: A,Z3: A,N: nat] :
( ( H
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H ) @ N ) @ ( power_power @ A @ Z3 @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
= ( times_times @ A @ H
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q5: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H ) @ Q5 ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ P5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_2177_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A7: A,N5: nat] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_2178_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N5: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I: A] : ( plus_plus @ A @ I @ ( one_one @ A ) )
@ N5
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_2179_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_2180_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_2181_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_2182_lessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_lessThan @ A @ X )
= ( set_ord_lessThan @ A @ Y ) )
= ( X = Y ) ) ) ).
% lessThan_eq_iff
thf(fact_2183_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K2: A] :
( ( member @ A @ I2 @ ( set_ord_lessThan @ A @ K2 ) )
= ( ord_less @ A @ I2 @ K2 ) ) ) ).
% lessThan_iff
thf(fact_2184_finite__lessThan,axiom,
! [K2: nat] : ( finite_finite2 @ nat @ ( set_ord_lessThan @ nat @ K2 ) ) ).
% finite_lessThan
thf(fact_2185_concat__bit__0,axiom,
! [K2: int,L: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K2 @ L )
= L ) ).
% concat_bit_0
thf(fact_2186_card__lessThan,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_lessThan @ nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_2187_dbl__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% dbl_simps(2)
thf(fact_2188_prod__zero__iff,axiom,
! [A: $tType,B: $tType] :
( ( semidom @ A )
=> ! [A6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 )
= ( zero_zero @ A ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ( F3 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_2189_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% lessThan_subset_iff
thf(fact_2190_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_2191_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_2192_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ~ ( member @ B @ X @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A6 ) )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 ) ) ) ) ) ) ).
% prod.insert
thf(fact_2193_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_2194_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K2: A] :
( ( minus_minus @ ( set @ A ) @ ( insert @ A @ K2 @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K2 ) )
= ( insert @ A @ K2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_2195_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_2196_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_2197_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( G3 @ ( minus_minus @ nat @ N @ ( suc @ I ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_2198_lessThan__non__empty,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
( ( set_ord_lessThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% lessThan_non_empty
thf(fact_2199_infinite__Iio,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_lessThan @ A @ A3 ) ) ) ).
% infinite_Iio
thf(fact_2200_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > A,H: B > A,A6: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( G3 @ X4 ) @ ( H @ X4 ) )
@ A6 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ H @ A6 ) ) ) ) ).
% prod.distrib
thf(fact_2201_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X4: A] : ( ord_less @ A @ X4 @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_2202_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
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_2203_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_2204_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_2205_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A6: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) ) ) ) ).
% prod_nonneg
thf(fact_2206_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A6: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) ) ) ) ).
% prod_pos
thf(fact_2207_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A6: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) ) ) ) ).
% prod_ge_1
thf(fact_2208_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ? [X5: B] :
( ( member @ B @ X5 @ A6 )
& ( ( F3 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_2209_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F3: nat > A,A3: nat,B2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A7: nat] : ( times_times @ A @ ( F3 @ A7 ) )
@ A3
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_2210_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_2211_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M2 ) @ ( set_ord_lessThan @ A @ N ) )
= ( ord_less @ A @ M2 @ N ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_2212_lessThan__Suc,axiom,
! [K2: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K2 ) )
= ( insert @ nat @ K2 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ).
% lessThan_Suc
thf(fact_2213_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_2214_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X4: A] : ( plus_plus @ A @ X4 @ X4 ) ) ) ) ).
% dbl_def
thf(fact_2215_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
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_2216_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
@ ^ [I: nat] : ( G3 @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_2217_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A6: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_2218_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R2: A > A > $o,S3: set @ B,H: B > A,G3: B > A] :
( ( R2 @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X15: A,Y15: A,X23: A,Y23: A] :
( ( ( R2 @ X15 @ X23 )
& ( R2 @ Y15 @ Y23 ) )
=> ( R2 @ ( times_times @ A @ X15 @ Y15 ) @ ( times_times @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( R2 @ ( H @ X3 ) @ ( G3 @ X3 ) ) )
=> ( R2 @ ( groups7121269368397514597t_prod @ B @ A @ H @ S3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ S3 ) ) ) ) ) ) ) ).
% prod.related
thf(fact_2219_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( ( member @ B @ X @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A6 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 ) ) )
& ( ~ ( member @ B @ X @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A6 ) )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_2220_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_2221_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,N: nat,I2: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N ) @ I2 )
= ( semiri8178284476397505188at_aux @ A @ Inc @ N @ ( Inc @ I2 ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_2222_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,I2: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I2 )
= I2 ) ) ).
% of_nat_aux.simps(1)
thf(fact_2223_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
@ ^ [I: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_2224_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( G3 @ ( minus_minus @ nat @ N @ ( suc @ I ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_2225_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P2: A > nat,N: A] :
( ! [X3: A] : ( ord_less_eq @ nat @ ( Q @ X3 ) @ ( P2 @ X3 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P2 @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( minus_minus @ nat @ ( P2 @ X4 ) @ ( Q @ X4 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_2226_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,I2: A,F3: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( member @ A @ I2 @ I5 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I2 ) )
=> ( ! [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_2227_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B6: set @ B,A6: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A6 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A6 @ B6 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B6 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_2228_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,B6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B6 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_2229_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,G3: B > A,B6: set @ B] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A6 @ B6 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_2230_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_2231_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_2232_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_2233_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,K2: A] :
( ( ( ord_less @ A @ X @ K2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X @ K2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_2234_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
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_2235_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
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_2236_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ M2 ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_2237_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N5 ) ) @ ( F3 @ N5 ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F3 @ M2 ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_2238_sumr__diff__mult__const2,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [F3: nat > A,N: nat,R3: A] :
( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ R3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( minus_minus @ A @ ( F3 @ I ) @ R3 )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sumr_diff_mult_const2
thf(fact_2239_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_2240_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_2241_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( member @ B @ X @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_2242_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,G3: B > A,X: B] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A6 ) )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_2243_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,B6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_2244_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,B6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ( inf_inf @ ( set @ B ) @ A6 @ B6 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_2245_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_2246_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_2247_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_2248_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_2249_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,B6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A6 @ B6 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ B6 @ A6 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_2250_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A,P: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_2251_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F4: nat > A > A,A7: nat,B5: nat,Acc2: A] : ( if @ A @ ( ord_less @ nat @ B5 @ A7 ) @ Acc2 @ ( set_fo6178422350223883121st_nat @ A @ F4 @ ( plus_plus @ nat @ A7 @ ( one_one @ nat ) ) @ B5 @ ( F4 @ A7 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_2252_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa2: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa2 @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa2 )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa2 @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_2253_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 @ ( insert @ 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 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_2254_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_2255_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_2256_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N: nat] :
( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_2257_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B6: set @ A,A6: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B6 @ 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 @ B6 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_2258_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_2259_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A6: set @ B,B6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) )
=> ( ( F3 @ X3 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ B6 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_2260_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 @ ( insert @ 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 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F3 @ A6 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_2261_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_2262_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_2263_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_2264_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_2265_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_2266_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_2267_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_2268_lemma__termdiff1,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [Z3: A,H: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H ) @ ( minus_minus @ nat @ M2 @ P5 ) ) @ ( power_power @ A @ Z3 @ P5 ) ) @ ( power_power @ A @ Z3 @ M2 ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( times_times @ A @ ( power_power @ A @ Z3 @ P5 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H ) @ ( minus_minus @ nat @ M2 @ P5 ) ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ M2 @ P5 ) ) ) )
@ ( set_ord_lessThan @ nat @ M2 ) ) ) ) ).
% lemma_termdiff1
thf(fact_2269_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( times_times @ A @ ( power_power @ A @ X @ P5 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P5 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_2270_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I ) ) ) @ ( power_power @ A @ X @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_2271_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,F3: nat > A,K5: A,K2: nat] :
( ! [P7: nat] :
( ( ord_less @ nat @ P7 @ N )
=> ( ord_less_eq @ A @ ( F3 @ P7 ) @ K5 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K5 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ K2 ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ K5 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_2272_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
@ ^ [I: nat] : ( times_times @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_2273_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
@ ^ [A7: nat] : ( plus_plus @ A @ ( F3 @ A7 ) )
@ A3
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_2274_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
@ ^ [I: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_2275_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_2276_sum__split__even__odd,axiom,
! [F3: nat > real,G3: nat > real,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) @ ( F3 @ I ) @ ( G3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( F3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( G3 @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) @ ( one_one @ nat ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_2277_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_2278_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_2279_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2280_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_2281_binomial__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( binomial @ N @ K2 )
= ( zero_zero @ nat ) )
= ( ord_less @ nat @ N @ K2 ) ) ).
% binomial_eq_0_iff
thf(fact_2282_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_2283_binomial__0__Suc,axiom,
! [K2: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K2 ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_2284_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_2285_prod__pos__nat__iff,axiom,
! [A: $tType,A6: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F3 @ A6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X4 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_2286_prod__int__eq,axiom,
! [I2: nat,J2: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I2 @ J2 ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X4: int] : X4
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I2 ) @ ( semiring_1_of_nat @ int @ J2 ) ) ) ) ).
% prod_int_eq
thf(fact_2287_prod__int__plus__eq,axiom,
! [I2: nat,J2: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I2 @ ( plus_plus @ nat @ I2 @ J2 ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X4: int] : X4
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I2 ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I2 @ J2 ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_2288_binomial__eq__0,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( ( binomial @ N @ K2 )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2289_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_2290_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_2291_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_2292_choose__mult__lemma,axiom,
! [M2: nat,R3: nat,K2: nat] :
( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R3 ) @ K2 ) @ ( plus_plus @ nat @ M2 @ K2 ) ) @ ( binomial @ ( plus_plus @ nat @ M2 @ K2 ) @ K2 ) )
= ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R3 ) @ K2 ) @ K2 ) @ ( binomial @ ( plus_plus @ nat @ M2 @ R3 ) @ M2 ) ) ) ).
% choose_mult_lemma
thf(fact_2293_binomial__le__pow,axiom,
! [R3: nat,N: nat] :
( ( ord_less_eq @ nat @ R3 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ R3 ) @ ( power_power @ nat @ N @ R3 ) ) ) ).
% binomial_le_pow
thf(fact_2294_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_2295_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_2296_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_2297_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_2298_binomial__absorb__comp,axiom,
! [N: nat,K2: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ N @ K2 ) @ ( binomial @ N @ K2 ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K2 ) ) ) ).
% binomial_absorb_comp
thf(fact_2299_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_2300_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_2301_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_2302_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_2303_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_2304_times__binomial__minus1__eq,axiom,
! [K2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( times_times @ nat @ K2 @ ( binomial @ N @ K2 ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_2305_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_2306_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
@ ^ [I: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_2307_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
@ ^ [I: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_2308_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_2309_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R3: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ R3 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R3 @ ( 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 @ R3 @ ( suc @ M2 ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_2310_vebt__buildup_Opelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_2311_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_2312_atMost__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_atMost @ A @ X )
= ( set_ord_atMost @ A @ Y ) )
= ( X = Y ) ) ) ).
% atMost_eq_iff
thf(fact_2313_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K2: A] :
( ( member @ A @ I2 @ ( set_ord_atMost @ A @ K2 ) )
= ( ord_less_eq @ A @ I2 @ K2 ) ) ) ).
% atMost_iff
thf(fact_2314_of__bool__less__eq__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P2: $o,Q: $o] :
( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( P2
=> Q ) ) ) ).
% of_bool_less_eq_iff
thf(fact_2315_of__bool__eq_I1_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A @ $false )
= ( zero_zero @ A ) ) ) ).
% of_bool_eq(1)
thf(fact_2316_of__bool__eq__0__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P2: $o] :
( ( ( zero_neq_one_of_bool @ A @ P2 )
= ( zero_zero @ A ) )
= ~ P2 ) ) ).
% of_bool_eq_0_iff
thf(fact_2317_of__bool__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P2: $o,Q: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( ~ P2
& Q ) ) ) ).
% of_bool_less_iff
thf(fact_2318_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P2: $o] :
( ( ( zero_neq_one_of_bool @ A @ P2 )
= ( one_one @ A ) )
= P2 ) ) ).
% of_bool_eq_1_iff
thf(fact_2319_of__bool__eq_I2_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A @ $true )
= ( one_one @ A ) ) ) ).
% of_bool_eq(2)
thf(fact_2320_of__nat__of__bool,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P2: $o] :
( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P2 ) )
= ( zero_neq_one_of_bool @ A @ P2 ) ) ) ).
% of_nat_of_bool
thf(fact_2321_of__bool__or__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P2
| Q ) )
= ( ord_max @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_or_iff
thf(fact_2322_finite__atMost,axiom,
! [K2: nat] : ( finite_finite2 @ nat @ ( set_ord_atMost @ nat @ K2 ) ) ).
% finite_atMost
thf(fact_2323_zero__less__of__bool__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P2 ) )
= P2 ) ) ).
% zero_less_of_bool_iff
thf(fact_2324_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% atMost_subset_iff
thf(fact_2325_of__bool__less__one__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( one_one @ A ) )
= ~ P2 ) ) ).
% of_bool_less_one_iff
thf(fact_2326_of__bool__not__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P2: $o] :
( ( zero_neq_one_of_bool @ A @ ~ P2 )
= ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P2 ) ) ) ) ).
% of_bool_not_iff
thf(fact_2327_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_2328_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_2329_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_2330_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_2331_card__atMost,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_2332_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_2333_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_2334_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_2335_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_2336_distinct__swap,axiom,
! [A: $tType,I2: nat,Xs2: list @ A,J2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs2 @ I2 @ ( nth @ A @ Xs2 @ J2 ) ) @ J2 @ ( nth @ A @ Xs2 @ I2 ) ) )
= ( distinct @ A @ Xs2 ) ) ) ) ).
% distinct_swap
thf(fact_2337_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A6: set @ B,F3: B > A,P2: B > $o] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ ( zero_neq_one_of_bool @ A @ ( P2 @ X4 ) ) )
@ A6 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P2 ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_2338_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A6: set @ B,P2: B > $o,F3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P2 @ X4 ) ) @ ( F3 @ X4 ) )
@ A6 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P2 ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_2339_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_2340_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_2341_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_2342_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_2343_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_2344_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o,Q2: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( zero_neq_one_of_bool @ A @ Q2 ) )
= ( P = Q2 ) ) ) ).
% of_bool_eq_iff
thf(fact_2345_of__bool__conj,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P2: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P2
& Q ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_conj
thf(fact_2346_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_2347_infinite__Iic,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_atMost @ A @ A3 ) ) ) ).
% infinite_Iic
thf(fact_2348_not__Iic__eq__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H3: A,L: A,H: A] :
( ( set_ord_atMost @ A @ H3 )
!= ( set_or1337092689740270186AtMost @ A @ L @ H ) ) ) ).
% not_Iic_eq_Icc
thf(fact_2349_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_2350_zero__less__eq__of__bool,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P2 ) ) ) ).
% zero_less_eq_of_bool
thf(fact_2351_of__bool__less__eq__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( one_one @ A ) ) ) ).
% of_bool_less_eq_one
thf(fact_2352_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P5: $o] : ( if @ A @ P5 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_2353_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P2: A > $o,P: $o] :
( ( P2 @ ( zero_neq_one_of_bool @ A @ P ) )
= ( ( P
=> ( P2 @ ( one_one @ A ) ) )
& ( ~ P
=> ( P2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_2354_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P2: A > $o,P: $o] :
( ( P2 @ ( zero_neq_one_of_bool @ A @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ ( one_one @ A ) ) )
| ( ~ P
& ~ ( P2 @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_2355_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_2356_lessThan__Suc__atMost,axiom,
! [K2: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K2 ) )
= ( set_ord_atMost @ nat @ K2 ) ) ).
% lessThan_Suc_atMost
thf(fact_2357_atMost__Suc,axiom,
! [K2: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K2 ) )
= ( insert @ nat @ ( suc @ K2 ) @ ( set_ord_atMost @ nat @ K2 ) ) ) ).
% atMost_Suc
thf(fact_2358_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_2359_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs2: list @ A,I2: nat,J2: nat] :
( ( distinct @ A @ Xs2 )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ Xs2 @ I2 )
= ( nth @ A @ Xs2 @ J2 ) )
= ( I2 = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_2360_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs: list @ A] :
! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ! [J: nat] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( I != J )
=> ( ( nth @ A @ Xs @ I )
!= ( nth @ A @ Xs @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_2361_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_2362_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_2363_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_2364_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_2365_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_2366_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_2367_distinct__Ex1,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ( distinct @ A @ Xs2 )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ? [X3: nat] :
( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( nth @ A @ Xs2 @ X3 )
= X )
& ! [Y6: nat] :
( ( ( ord_less @ nat @ Y6 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( nth @ A @ Xs2 @ Y6 )
= X ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_2368_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A3 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_2369_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_2370_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K2: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( gbinomial @ A @ A3 @ K2 ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_2371_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_2372_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_2373_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_2374_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
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_2375_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,I2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( minus_minus @ A @ ( F3 @ I ) @ ( F3 @ ( suc @ I ) ) )
@ ( set_ord_atMost @ nat @ I2 ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ ( suc @ I2 ) ) ) ) ) ).
% sum_telescope
thf(fact_2376_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
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ X4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( D3 @ I ) @ ( power_power @ A @ X4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
= ( ! [I: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( C3 @ I )
= ( D3 @ I ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_2377_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
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_2378_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I ) @ ( set_ord_lessThan @ nat @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( A3 @ I @ J )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_2379_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_2380_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [U: A] :
( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_ord_atMost @ A @ U ) ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_2381_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I ) @ ( set_ord_lessThan @ nat @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( A3 @ I @ J )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_2382_sum__choose__lower,axiom,
! [R3: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R3 @ K3 ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R3 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_2383_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_2384_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_2385_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_2386_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C3: nat > A,N: nat,K2: nat] :
( ! [W: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ W @ I ) )
@ ( 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_2387_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ X4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) )
= ( ! [I: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( C3 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_2388_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
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_2389_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_2390_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
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_2391_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_2392_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_2393_vandermonde,axiom,
! [M2: nat,N: nat,R3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M2 @ K3 ) @ ( binomial @ N @ ( minus_minus @ nat @ R3 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R3 ) )
= ( binomial @ ( plus_plus @ nat @ M2 @ N ) @ R3 ) ) ).
% vandermonde
thf(fact_2394_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P2: A > $o,A3: A] :
( ! [A5: A] :
( ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A5 )
=> ( P2 @ A5 ) )
=> ( ! [A5: A,B4: $o] :
( ( P2 @ 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 )
=> ( P2 @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B4 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) ) ) )
=> ( P2 @ A3 ) ) ) ) ).
% bits_induct
thf(fact_2395_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_2396_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_2397_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_2398_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_2399_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
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ X4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I: nat] :
( ( ord_less_eq @ nat @ I @ N )
& ( ( C3 @ I )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_2400_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
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ Z4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_2401_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
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ Z4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ).
% polyfun_roots_card
thf(fact_2402_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: nat > A,A3: A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ A3 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ~ ! [B4: nat > A] :
~ ! [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ Z5 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ Z5 @ A3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( B4 @ I ) @ ( power_power @ A @ Z5 @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_2403_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: nat > A,N: nat,A3: A] :
? [B4: nat > A] :
! [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ Z5 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A
@ ( times_times @ A @ ( minus_minus @ A @ Z5 @ A3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( B4 @ I ) @ ( power_power @ A @ Z5 @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ A3 @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% polyfun_linear_factor
thf(fact_2404_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_2405_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_2406_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_2407_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_2408_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_2409_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
@ ^ [I: nat] : ( plus_plus @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_2410_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
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ Z4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ Z4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_2411_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M2: nat,A3: nat > A,N: nat,B2: nat > A,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ M2 @ I3 )
=> ( ( A3 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [J3: nat] :
( ( ord_less @ nat @ N @ J3 )
=> ( ( B2 @ J3 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A3 @ I ) @ ( power_power @ A @ X @ I ) )
@ ( set_ord_atMost @ nat @ M2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J: nat] : ( times_times @ A @ ( B2 @ J ) @ ( power_power @ A @ X @ J ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [R: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A3 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R ) )
@ ( power_power @ A @ X @ R ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_2412_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
@ ^ [I: nat] : ( times_times @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_2413_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat,K2: A] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ X4 @ I ) )
@ ( 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_2414_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_2415_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_2416_polynomial__product__nat,axiom,
! [M2: nat,A3: nat > nat,N: nat,B2: nat > nat,X: nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ M2 @ I3 )
=> ( ( A3 @ I3 )
= ( zero_zero @ nat ) ) )
=> ( ! [J3: nat] :
( ( ord_less @ nat @ N @ J3 )
=> ( ( B2 @ J3 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ ( A3 @ I ) @ ( power_power @ nat @ X @ I ) )
@ ( set_ord_atMost @ nat @ M2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J: nat] : ( times_times @ nat @ ( B2 @ J ) @ ( power_power @ nat @ X @ J ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [R: nat] :
( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( A3 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R ) )
@ ( power_power @ nat @ X @ R ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_2417_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_2418_set__update__distinct,axiom,
! [A: $tType,Xs2: list @ A,N: nat,X: A] :
( ( distinct @ A @ Xs2 )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs2 @ N @ X ) )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert @ A @ ( nth @ A @ Xs2 @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_2419_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P: nat,K2: nat,G3: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P )
=> ( ( ord_less_eq @ nat @ K2 @ P )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J: nat] : ( if @ A @ ( ord_less @ nat @ J @ K2 ) @ ( G3 @ J ) @ ( if @ A @ ( J = K2 ) @ ( zero_zero @ A ) @ ( H @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J: nat] : ( if @ A @ ( ord_less @ nat @ J @ K2 ) @ ( G3 @ J ) @ ( H @ J ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_2420_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P: nat,K2: nat,G3: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P )
=> ( ( ord_less_eq @ nat @ K2 @ P )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J: nat] : ( if @ A @ ( ord_less @ nat @ J @ K2 ) @ ( G3 @ J ) @ ( if @ A @ ( J = K2 ) @ ( one_one @ A ) @ ( H @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J: nat] : ( if @ A @ ( ord_less @ nat @ J @ K2 ) @ ( G3 @ J ) @ ( H @ J ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_2421_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J ) @ 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_2422_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_2423_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( gbinomial @ A @ A3 @ K2 )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_2424_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp0
thf(fact_2425_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A3: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A3 @ I ) @ ( power_power @ A @ X @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A3 @ I ) @ ( power_power @ A @ Y @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( A3 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J @ K3 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y @ K3 ) ) @ ( power_power @ A @ X @ J ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ J ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_2426_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_2427_choose__linear__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ I @ ( binomial @ N @ I ) )
@ ( 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_2428_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_2429_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
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ Z5 @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) )
@ ( times_times @ real @ E3 @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_2430_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A3: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A3 @ I ) @ ( power_power @ A @ X @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A3 @ I ) @ ( power_power @ A @ Y @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A3 @ I ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J ) @ N ) )
@ ( power_power @ A @ X @ J ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_2431_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_2432_Divides_Oadjust__div__eq,axiom,
! [Q2: int,R3: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q2 @ R3 ) )
= ( plus_plus @ int @ Q2
@ ( zero_neq_one_of_bool @ int
@ ( R3
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_2433_list__decode_Opinduct,axiom,
! [A0: nat,P2: nat > $o] :
( ( accp @ nat @ nat_list_decode_rel @ A0 )
=> ( ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( P2 @ ( zero_zero @ nat ) ) )
=> ( ! [N3: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N3 ) )
=> ( ! [X5: nat,Y6: nat] :
( ( ( product_Pair @ nat @ nat @ X5 @ Y6 )
= ( nat_prod_decode @ N3 ) )
=> ( P2 @ Y6 ) )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_2434_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A7: A,K3: nat] :
( if @ A
@ ( K3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A7 @ ( semiring_1_of_nat @ A @ L2 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_2435_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_2436_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_2437_set__decode__0,axiom,
! [X: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% set_decode_0
thf(fact_2438_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_2439_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_2440_of__nat__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( ^ [N5: nat] : N5 ) ) ).
% of_nat_id
thf(fact_2441_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_2442_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_2443_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_2444_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_2445_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_2446_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_2447_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_2448_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_2449_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_2450_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_right
thf(fact_2451_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( times_times @ A @ A3 @ B2 ) ) ) ).
% minus_mult_minus
thf(fact_2452_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_left
thf(fact_2453_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_2454_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_2455_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_2456_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_2457_dvd__minus__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ X @ ( uminus_uminus @ A @ Y ) )
= ( dvd_dvd @ A @ X @ Y ) ) ) ).
% dvd_minus_iff
thf(fact_2458_minus__dvd__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X ) @ Y )
= ( dvd_dvd @ A @ X @ Y ) ) ) ).
% minus_dvd_iff
thf(fact_2459_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ int @ M2 ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M2
= ( zero_zero @ nat ) ) ) ) ).
% negative_eq_positive
thf(fact_2460_signed__take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% signed_take_bit_of_0
thf(fact_2461_set__decode__inverse,axiom,
! [N: nat] :
( ( nat_set_encode @ ( nat_set_decode @ N ) )
= N ) ).
% set_decode_inverse
thf(fact_2462_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_2463_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_2464_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_2465_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_2466_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_2467_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_2468_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_2469_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_2470_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_2471_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_2472_verit__minus__simplify_I3_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( minus_minus @ B @ ( zero_zero @ B ) @ B2 )
= ( uminus_uminus @ B @ B2 ) ) ) ).
% verit_minus_simplify(3)
thf(fact_2473_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_2474_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_2475_mult__minus1__right,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: A] :
( ( times_times @ A @ Z3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ Z3 ) ) ) ).
% mult_minus1_right
thf(fact_2476_mult__minus1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z3 )
= ( uminus_uminus @ A @ Z3 ) ) ) ).
% mult_minus1
thf(fact_2477_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_2478_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_2479_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_2480_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_2481_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_2482_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_2483_dbl__inc__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% dbl_inc_simps(4)
thf(fact_2484_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_2485_set__encode__inverse,axiom,
! [A6: set @ nat] :
( ( finite_finite2 @ nat @ A6 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A6 ) )
= A6 ) ) ).
% set_encode_inverse
thf(fact_2486_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_2487_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_2488_diff__numeral__special_I12_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(12)
thf(fact_2489_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_2490_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_2491_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_2492_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_2493_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_2494_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(2)
thf(fact_2495_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ A @ V3 ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(3)
thf(fact_2496_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(4)
thf(fact_2497_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_2498_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_2499_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_2500_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_2501_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_2502_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_2503_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_2504_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V3: num,W2: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V3 @ W2 ) ) @ Y ) ) ) ).
% semiring_norm(172)
thf(fact_2505_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V3: num,W2: num,Y: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V3 ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V3 @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(171)
thf(fact_2506_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V3: num,W2: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V3 @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(170)
thf(fact_2507_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_2508_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_2509_dbl__dec__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( zero_zero @ A ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% dbl_dec_simps(2)
thf(fact_2510_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_2511_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_2512_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_2513_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= A3 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_2514_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_2515_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_2516_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_2517_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_2518_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_2519_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_2520_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_2521_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_2522_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_2523_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_2524_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_2525_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_2526_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_2527_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_2528_set__decode__Suc,axiom,
! [N: nat,X: nat] :
( ( member @ nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
= ( member @ nat @ N @ ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_2529_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_2530_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_2531_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_2532_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_2533_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_2534_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_2535_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_self
thf(fact_2536_fact__nonzero,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ N )
!= ( zero_zero @ A ) ) ) ).
% fact_nonzero
thf(fact_2537_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_2538_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_2539_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_2540_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_2541_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_2542_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_2543_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_2544_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_2545_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_2546_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_2547_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_mult_commute
thf(fact_2548_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ A3 )
= ( times_times @ A @ B2 @ B2 ) )
= ( ( A3 = B2 )
| ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% square_eq_iff
thf(fact_2549_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_2550_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_2551_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_2552_one__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ( ( one_one @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% one_neq_neg_one
thf(fact_2553_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_2554_signed__take__bit__mult,axiom,
! [N: nat,K2: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ K2 @ L ) ) ) ).
% signed_take_bit_mult
thf(fact_2555_fact__less__mono__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M2 ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_2556_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_2557_fact__not__neg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] :
~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% fact_not_neg
thf(fact_2558_fact__gt__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_gt_zero
thf(fact_2559_finite__set__decode,axiom,
! [N: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N ) ) ).
% finite_set_decode
thf(fact_2560_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_2561_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_2562_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_2563_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_2564_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_2565_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_2566_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_2567_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_2568_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_2569_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_2570_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_2571_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_2572_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_2573_zero__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% zero_neq_neg_one
thf(fact_2574_less__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% less_minus_one_simps(4)
thf(fact_2575_less__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% less_minus_one_simps(2)
thf(fact_2576_numeral__times__minus__swap,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [W2: num,X: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ ( uminus_uminus @ A @ X ) )
= ( times_times @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_2577_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_2578_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_2579_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_2580_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_2581_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [X: A] :
( ( ( times_times @ A @ X @ X )
= ( one_one @ A ) )
= ( ( X
= ( one_one @ A ) )
| ( X
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_2582_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_2583_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A7: A,B5: A] : ( plus_plus @ A @ A7 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2584_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A7: A,B5: A] : ( plus_plus @ A @ A7 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_2585_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B6: A,K2: A,B2: A,A3: A] :
( ( B6
= ( plus_plus @ A @ K2 @ B2 ) )
=> ( ( minus_minus @ A @ A3 @ B6 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K2 ) @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_2586_dvd__neg__div,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% dvd_neg_div
thf(fact_2587_dvd__div__neg,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% dvd_div_neg
thf(fact_2588_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_2589_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( times_times @ real @ U @ U ) ) @ ( times_times @ real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_2590_int__of__nat__induct,axiom,
! [P2: int > $o,Z3: int] :
( ! [N3: nat] : ( P2 @ ( semiring_1_of_nat @ int @ N3 ) )
=> ( ! [N3: nat] : ( P2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) )
=> ( P2 @ Z3 ) ) ) ).
% int_of_nat_induct
thf(fact_2591_int__cases,axiom,
! [Z3: int] :
( ! [N3: nat] :
( Z3
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( Z3
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_2592_zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ( times_times @ int @ M2 @ N )
= ( one_one @ int ) )
= ( ( ( M2
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) )
| ( ( M2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
& ( N
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_2593_pos__zmult__eq__1__iff__lemma,axiom,
! [M2: int,N: int] :
( ( ( times_times @ int @ M2 @ N )
= ( one_one @ int ) )
=> ( ( M2
= ( one_one @ int ) )
| ( M2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_2594_zmod__zminus1__not__zero,axiom,
! [K2: int,L: int] :
( ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K2 ) @ L )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K2 @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus1_not_zero
thf(fact_2595_zmod__zminus2__not__zero,axiom,
! [K2: int,L: int] :
( ( ( modulo_modulo @ int @ K2 @ ( uminus_uminus @ int @ L ) )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K2 @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus2_not_zero
thf(fact_2596_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_2597_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_2598_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_2599_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_2600_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_2601_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_2602_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_2603_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_2604_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_2605_less__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% less_minus_one_simps(1)
thf(fact_2606_less__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% less_minus_one_simps(3)
thf(fact_2607_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_2608_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_2609_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_2610_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_2611_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_2612_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A7: 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 @ A7 ) @ K3 ) ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_2613_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_2614_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_2615_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_2616_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_2617_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_2618_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3
= ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C3 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_2619_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) )
= A3 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B2 )
= ( times_times @ A @ A3 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_2620_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= C3 )
= ( ( uminus_uminus @ A @ A3 )
= ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_2621_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( C3
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) )
= ( ( times_times @ A @ C3 @ B2 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_2622_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_2623_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_2624_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_2625_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_2626_fact__less__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ A @ ( semiring_char_0_fact @ A @ M2 ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ) ).
% fact_less_mono
thf(fact_2627_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_2628_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% inf_shunt
thf(fact_2629_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_2630_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_2631_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P: A,Q2: A,R3: A] :
( ( ord_less_eq @ A @ P @ ( sup_sup @ A @ Q2 @ R3 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P @ ( uminus_uminus @ A @ Q2 ) ) @ R3 ) ) ) ).
% sup_neg_inf
thf(fact_2632_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ ( uminus_uminus @ A @ Y ) ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) ) ) ) ).
% shunt2
thf(fact_2633_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y ) @ Z3 ) ) ) ) ).
% shunt1
thf(fact_2634_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_2635_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( M2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_2636_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M2 ) ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M2
= ( zero_zero @ nat ) ) ) ) ).
% int_zle_neg
thf(fact_2637_zmod__zminus1__eq__if,axiom,
! [A3: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( minus_minus @ int @ B2 @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_2638_zmod__zminus2__eq__if,axiom,
! [A3: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ B2 ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_2639_fact__div__fact__le__pow,axiom,
! [R3: nat,N: nat] :
( ( ord_less_eq @ nat @ R3 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ R3 ) ) ) @ ( power_power @ nat @ N @ R3 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_2640_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_2641_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_2642_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ 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 @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_2643_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ 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 @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_2644_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_2645_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_2646_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_2647_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_2648_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_2649_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C3 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_2650_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
= B2 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_2651_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z3 ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z3 ) ) @ Z3 ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_2652_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_2653_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_2654_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_2655_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z3 ) ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z3 ) ) @ Z3 ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_2656_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_2657_int__cases3,axiom,
! [K2: int] :
( ( K2
!= ( zero_zero @ int ) )
=> ( ! [N3: nat] :
( ( K2
= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% int_cases3
thf(fact_2658_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K2 )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_2659_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N: nat,K2: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K2 )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N @ K2 ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_2660_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
= ( zero_zero @ A ) )
= ( ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N )
& ( A3
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_2661_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_2662_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_2663_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N3: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_2664_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_2665_div__eq__minus1,axiom,
! [B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ).
% div_eq_minus1
thf(fact_2666_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,P2: B > $o,H: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G3 @ X4 ) )
@ A6 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P2 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A6 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P2 ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_2667_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_2668_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_2669_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_2670_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_2671_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_2672_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_2673_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_2674_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_2675_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_2676_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_2677_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% neg_int_cases
thf(fact_2678_minus__mod__int__eq,axiom,
! [L: int,K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K2 ) @ L )
= ( minus_minus @ int @ ( minus_minus @ int @ L @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K2 @ ( one_one @ int ) ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_2679_zmod__minus1,axiom,
! [B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
= ( minus_minus @ int @ B2 @ ( one_one @ int ) ) ) ) ).
% zmod_minus1
thf(fact_2680_zdiv__zminus1__eq__if,axiom,
! [B2: int,A3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus1_eq_if
thf(fact_2681_zdiv__zminus2__eq__if,axiom,
! [B2: int,A3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus2_eq_if
thf(fact_2682_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_2683_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N5: nat,A7: A] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_2684_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_2685_zminus1__lemma,axiom,
! [A3: int,B2: int,Q2: int,R3: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R3 ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A3 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R3
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q2 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q2 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R3 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_2686_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_2687_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_2688_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_2689_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_2690_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M6: nat] :
( if @ A
@ ( M6
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M6 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M6 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_2691_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N5: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_2692_fact__reduce,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% fact_reduce
thf(fact_2693_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R3: A,K2: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R3 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R3 ) @ K2 ) )
= ( times_times @ A @ R3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R3 ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_2694_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_2695_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A7: 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 ) @ A7 ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_2696_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_2697_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_2698_Bernoulli__inequality,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_2699_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_2700_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_2701_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_2702_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_2703_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_2704_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_2705_int__bit__induct,axiom,
! [P2: int > $o,K2: int] :
( ( P2 @ ( zero_zero @ int ) )
=> ( ( P2 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K: int] :
( ( P2 @ K )
=> ( ( K
!= ( zero_zero @ int ) )
=> ( P2 @ ( times_times @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K: int] :
( ( P2 @ K )
=> ( ( K
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P2 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P2 @ K2 ) ) ) ) ) ).
% int_bit_induct
thf(fact_2706_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_2707_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
@ ^ [I: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K2 ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_2708_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_2709_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,Z3: A,A3: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( ( power_power @ A @ Z3 @ N )
= A3 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] :
( times_times @ A
@ ( if @ A
@ ( I
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A3 )
@ ( if @ A @ ( I = N ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z3 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_2710_set__decode__plus__power__2,axiom,
! [N: nat,Z3: nat] :
( ~ ( member @ nat @ N @ ( nat_set_decode @ Z3 ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z3 ) )
= ( insert @ nat @ N @ ( nat_set_decode @ Z3 ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_2711_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I ) @ ( semiring_1_of_nat @ A @ I ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_2712_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X4: nat] :
( collect @ nat
@ ^ [N5: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_2713_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_2714_binomial__code,axiom,
( binomial
= ( ^ [N5: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N5 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N5 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N5 @ ( minus_minus @ nat @ N5 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N5 @ K3 ) @ ( one_one @ nat ) ) @ N5 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_2715_Maclaurin__lemma,axiom,
! [H: real,F3: real > real,J2: nat > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H )
=> ? [B9: real] :
( ( F3 @ H )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J2 @ M6 ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ H @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ B9 @ ( divide_divide @ real @ ( power_power @ real @ H @ N ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_2716_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: real,N: nat,Diff: nat > A > real] :
( ( X
= ( zero_zero @ real ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ X @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_2717_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N5: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N5 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_2718_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_2719_sin__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N5 ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X ) ) ).
% sin_paired
thf(fact_2720_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( member @ int @ K3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
@ ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
@ ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_2721_sin__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sin_zero
thf(fact_2722_bit_Oconj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_right
thf(fact_2723_bit_Oconj__zero__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_left
thf(fact_2724_zero__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_and_eq
thf(fact_2725_and__zero__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_zero_eq
thf(fact_2726_sin__coeff__0,axiom,
( ( sin_coeff @ ( zero_zero @ nat ) )
= ( zero_zero @ real ) ) ).
% sin_coeff_0
thf(fact_2727_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_2728_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_2729_and__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(3)
thf(fact_2730_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_2731_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_2732_and__int_Oelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa2 )
= Y )
=> ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_2733_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( sin_coeff @ M6 ) @ ( power_power @ real @ X @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_2734_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( sin_coeff @ M6 ) @ ( power_power @ real @ X @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_2735_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( cos_coeff @ M6 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M6 ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( one_one @ real ) ) ).
% sumr_cos_zero_one
thf(fact_2736_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_2737_Maclaurin__sin__expansion,axiom,
! [X: real,N: nat] :
? [T7: real] :
( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( sin_coeff @ M6 ) @ ( power_power @ real @ X @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_2738_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_2739_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_2740_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_2741_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_2742_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_2743_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_2744_and__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(3)
thf(fact_2745_zdiv__numeral__Bit1,axiom,
! [V3: num,W2: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit1 @ V3 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V3 ) @ ( numeral_numeral @ int @ W2 ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_2746_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_2747_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_2748_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_2749_sin__npi2,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_npi2
thf(fact_2750_sin__npi,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_npi
thf(fact_2751_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_2752_and__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(4)
thf(fact_2753_and__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(6)
thf(fact_2754_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_2755_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_2756_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_2757_Suc__div__eq__add3__div__numeral,axiom,
! [M2: nat,V3: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V3 ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V3 ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_2758_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_2759_Suc__mod__eq__add3__mod__numeral,axiom,
! [M2: nat,V3: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V3 ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V3 ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_2760_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_2761_sin__two__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_two_pi
thf(fact_2762_sin__2pi__minus,axiom,
! [X: real] :
( ( sin @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X ) )
= ( uminus_uminus @ real @ ( sin @ real @ X ) ) ) ).
% sin_2pi_minus
thf(fact_2763_sin__periodic,axiom,
! [X: real] :
( ( sin @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( sin @ real @ X ) ) ).
% sin_periodic
thf(fact_2764_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_2765_zmod__numeral__Bit1,axiom,
! [V3: num,W2: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V3 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V3 ) @ ( numeral_numeral @ int @ W2 ) ) ) @ ( one_one @ int ) ) ) ).
% zmod_numeral_Bit1
thf(fact_2766_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_2767_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_2768_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_2769_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one2 )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X32: num] :
( Y
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_2770_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_2771_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_2772_cong__exp__iff__simps_I13_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q2: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ Q2 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_2773_cong__exp__iff__simps_I12_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q2: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_2774_cong__exp__iff__simps_I10_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q2: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_2775_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_2776_power__numeral__odd,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z3: A,W2: num] :
( ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ ( bit1 @ W2 ) ) )
= ( times_times @ A @ ( times_times @ A @ Z3 @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) ) @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) ) ) ) ).
% power_numeral_odd
thf(fact_2777_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q2: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_2778_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_2779_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_2780_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_2781_num_Osize_I6_J,axiom,
! [X33: num] :
( ( size_size @ num @ ( bit1 @ X33 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X33 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(6)
thf(fact_2782_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q2: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_2783_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q2: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_2784_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_2785_card__3__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X4: A,Y4: A,Z4: A] :
( ( S3
= ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( insert @ A @ Z4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X4 != Y4 )
& ( Y4 != Z4 )
& ( X4 != Z4 ) ) ) ) ).
% card_3_iff
thf(fact_2786_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_2787_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_2788_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_2789_sin__pi__divide__n__ge__0,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% sin_pi_divide_n_ge_0
thf(fact_2790_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_2791_sin__lt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ pi @ X )
=> ( ( ord_less @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_lt_zero
thf(fact_2792_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M6: nat,N5: nat] :
( if @ nat
@ ( ( M6
= ( zero_zero @ nat ) )
| ( N5
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_2793_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M6: nat,N5: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M6 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_2794_sin__le__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ pi @ X )
=> ( ( ord_less @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_le_zero
thf(fact_2795_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_2796_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_2797_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_2798_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ? [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_2799_sin__zero__iff,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [N5: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N5 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N5: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N5 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_2800_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ? [T7: real] :
( ( ord_less @ real @ X @ T7 )
& ( ord_less @ real @ T7 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( cos_coeff @ M6 ) @ ( power_power @ real @ X @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_2801_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ X )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( cos_coeff @ M6 ) @ ( power_power @ real @ X @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_2802_Maclaurin__sin__expansion2,axiom,
! [X: real,N: nat] :
? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( sin_coeff @ M6 ) @ ( power_power @ real @ X @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_2803_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_2804_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_2805_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_2806_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_2807_abs__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_abs
thf(fact_2808_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_2809_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_2810_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_2811_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_2812_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_2813_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% abs_mult_self_eq
thf(fact_2814_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_2815_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_2816_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_2817_abs__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_minus
thf(fact_2818_dvd__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: A,K2: A] :
( ( dvd_dvd @ A @ M2 @ ( abs_abs @ A @ K2 ) )
= ( dvd_dvd @ A @ M2 @ K2 ) ) ) ).
% dvd_abs_iff
thf(fact_2819_abs__dvd__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: A,K2: A] :
( ( dvd_dvd @ A @ ( abs_abs @ A @ M2 ) @ K2 )
= ( dvd_dvd @ A @ M2 @ K2 ) ) ) ).
% abs_dvd_iff
thf(fact_2820_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_2821_abs__bool__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: $o] :
( ( abs_abs @ A @ ( zero_neq_one_of_bool @ A @ P2 ) )
= ( zero_neq_one_of_bool @ A @ P2 ) ) ) ).
% abs_bool_eq
thf(fact_2822_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_2823_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_2824_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_2825_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_2826_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_2827_abs__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( one_one @ A ) ) ) ).
% abs_neg_one
thf(fact_2828_cos__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% cos_zero
thf(fact_2829_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_2830_eq__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ( numeral_numeral @ nat @ K2 )
= ( suc @ N ) )
= ( ( pred_numeral @ K2 )
= N ) ) ).
% eq_numeral_Suc
thf(fact_2831_Suc__eq__numeral,axiom,
! [N: nat,K2: num] :
( ( ( suc @ N )
= ( numeral_numeral @ nat @ K2 ) )
= ( N
= ( pred_numeral @ K2 ) ) ) ).
% Suc_eq_numeral
thf(fact_2832_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
@ ^ [I: A] : ( abs_abs @ B @ ( F3 @ I ) )
@ A6 ) ) ) ).
% sum_abs
thf(fact_2833_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_2834_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_2835_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_2836_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_2837_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_2838_pred__numeral__simps_I3_J,axiom,
! [K2: num] :
( ( pred_numeral @ ( bit1 @ K2 ) )
= ( numeral_numeral @ nat @ ( bit0 @ K2 ) ) ) ).
% pred_numeral_simps(3)
thf(fact_2839_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_2840_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_2841_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_2842_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_2843_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_2844_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_2845_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
@ ^ [I: A] : ( abs_abs @ B @ ( F3 @ I ) )
@ A6 ) ) ) ).
% sum_abs_ge_zero
thf(fact_2846_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_2847_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_2848_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_2849_sin__cos__squared__add3,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ X ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ X ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add3
thf(fact_2850_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_2851_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_2852_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_2853_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_2854_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_2855_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_2856_cos__two__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_two_pi
thf(fact_2857_cos__periodic,axiom,
! [X: real] :
( ( cos @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cos @ real @ X ) ) ).
% cos_periodic
thf(fact_2858_cos__2pi__minus,axiom,
! [X: real] :
( ( cos @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X ) )
= ( cos @ real @ X ) ) ).
% cos_2pi_minus
thf(fact_2859_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_2860_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_2861_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_2862_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_2863_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_2864_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_2865_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_2866_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_2867_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_2868_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_2869_abs__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( ( abs_abs @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0_iff
thf(fact_2870_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_mult
thf(fact_2871_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_2872_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_2873_abs__eq__iff,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
( ( ( abs_abs @ A @ X )
= ( abs_abs @ A @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% abs_eq_iff
thf(fact_2874_dvd__if__abs__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [L: A,K2: A] :
( ( ( abs_abs @ A @ L )
= ( abs_abs @ A @ K2 ) )
=> ( dvd_dvd @ A @ L @ K2 ) ) ) ).
% dvd_if_abs_eq
thf(fact_2875_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_2876_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_2877_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_2878_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_2879_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C3: A,B2: A,D3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ C3 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D3 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C3 @ D3 ) ) ) ) ) ).
% abs_mult_less
thf(fact_2880_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_2881_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_2882_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_2883_nonzero__abs__divide,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% nonzero_abs_divide
thf(fact_2884_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_2885_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_2886_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_2887_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_2888_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ B2 )
= ( ( ord_less @ A @ A3 @ B2 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_less_iff
thf(fact_2889_polar__Ex,axiom,
! [X: real,Y: real] :
? [R4: real,A5: real] :
( ( X
= ( times_times @ real @ R4 @ ( cos @ real @ A5 ) ) )
& ( Y
= ( times_times @ real @ R4 @ ( sin @ real @ A5 ) ) ) ) ).
% polar_Ex
thf(fact_2890_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_2891_sin__cos__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( plus_plus @ real @ ( times_times @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) ) @ ( times_times @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) @ ( one_one @ real ) ) ).
% sin_cos_le1
thf(fact_2892_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E2 ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_2893_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X )
= ( abs_abs @ A @ ( times_times @ A @ Y @ X ) ) ) ) ) ).
% abs_mult_pos
thf(fact_2894_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_2895_cos__one__sin__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
= ( one_one @ A ) )
=> ( ( sin @ A @ X )
= ( zero_zero @ A ) ) ) ) ).
% cos_one_sin_zero
thf(fact_2896_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_2897_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_2898_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_2899_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X ) @ Y )
= ( abs_abs @ A @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% abs_div_pos
thf(fact_2900_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_2901_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A7: A] : ( if @ A @ ( ord_less @ A @ A7 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A7 ) @ A7 ) ) ) ) ).
% abs_if_raw
thf(fact_2902_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A7: A] : ( if @ A @ ( ord_less @ A @ A7 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A7 ) @ A7 ) ) ) ) ).
% abs_if
thf(fact_2903_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_2904_sin__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( sin @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sin @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% sin_add
thf(fact_2905_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R3 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R3 ) @ X )
& ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A3 @ R3 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_2906_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_2907_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_2908_sin__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( sin @ A @ ( minus_minus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( times_times @ A @ ( sin @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% sin_diff
thf(fact_2909_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R3 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ A3 @ R3 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_2910_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_2911_lessThan__nat__numeral,axiom,
! [K2: num] :
( ( set_ord_lessThan @ nat @ ( numeral_numeral @ nat @ K2 ) )
= ( insert @ nat @ ( pred_numeral @ K2 ) @ ( set_ord_lessThan @ nat @ ( pred_numeral @ K2 ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_2912_atMost__nat__numeral,axiom,
! [K2: num] :
( ( set_ord_atMost @ nat @ ( numeral_numeral @ nat @ K2 ) )
= ( insert @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( set_ord_atMost @ nat @ ( pred_numeral @ K2 ) ) ) ) ).
% atMost_nat_numeral
thf(fact_2913_cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cos @ A @ ( plus_plus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% cos_add
thf(fact_2914_cos__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cos @ A @ ( minus_minus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% cos_diff
thf(fact_2915_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_2916_sin__zero__norm__cos__one,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( cos @ A @ X ) )
= ( one_one @ real ) ) ) ) ).
% sin_zero_norm_cos_one
thf(fact_2917_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_2918_sin__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ X ) ) @ ( cos @ A @ X ) ) ) ) ).
% sin_double
thf(fact_2919_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( abs_abs @ A @ Y ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_2920_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M6: num,N5: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M6 ) @ ( numeral_numeral @ int @ N5 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M6 ) @ ( numeral_numeral @ int @ N5 ) ) ) ) ) ).
% divmod_int_def
thf(fact_2921_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: A > A > $o,X: A] :
( ! [X3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( P2 @ X3 @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P2 @ ( abs_abs @ A @ X ) @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_2922_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ Y ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_2923_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_2924_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M6: num,N5: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M6 ) @ ( numeral_numeral @ A @ N5 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M6 ) @ ( numeral_numeral @ A @ N5 ) ) ) ) ) ) ).
% divmod_def
thf(fact_2925_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M6: num,N5: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M6 ) @ ( numeral_numeral @ nat @ N5 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M6 ) @ ( numeral_numeral @ nat @ N5 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_2926_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_2927_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,X: A > B,A3: A > B,B2: B,Delta: B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I5 )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I3 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I: A] : ( times_times @ B @ ( A3 @ I ) @ ( X @ I ) )
@ I5 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_2928_cos__plus__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( plus_plus @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_2929_cos__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( times_times @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( cos @ A @ ( minus_minus @ A @ W2 @ Z3 ) ) @ ( cos @ A @ ( plus_plus @ A @ W2 @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_cos
thf(fact_2930_cos__double__less__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) ) @ ( one_one @ real ) ) ) ) ).
% cos_double_less_one
thf(fact_2931_cos__double__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ W2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( cos @ A @ W2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) ) ) ) ).
% cos_double_cos
thf(fact_2932_cos__treble__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ X ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ ( cos @ A @ X ) ) ) ) ) ).
% cos_treble_cos
thf(fact_2933_cos__diff__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( minus_minus @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ Z3 @ W2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_2934_sin__diff__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( minus_minus @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_2935_sin__plus__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( plus_plus @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_2936_cos__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( times_times @ A @ ( cos @ A @ W2 ) @ ( sin @ A @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( sin @ A @ ( plus_plus @ A @ W2 @ Z3 ) ) @ ( sin @ A @ ( minus_minus @ A @ W2 @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_sin
thf(fact_2937_sin__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( times_times @ A @ ( sin @ A @ W2 ) @ ( cos @ A @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( sin @ A @ ( plus_plus @ A @ W2 @ Z3 ) ) @ ( sin @ A @ ( minus_minus @ A @ W2 @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_cos
thf(fact_2938_sin__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z3: A] :
( ( times_times @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( cos @ A @ ( minus_minus @ A @ W2 @ Z3 ) ) @ ( cos @ A @ ( plus_plus @ A @ W2 @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_sin
thf(fact_2939_cos__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_double
thf(fact_2940_Maclaurin__cos__expansion,axiom,
! [X: real,N: nat] :
? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( cos_coeff @ M6 ) @ ( power_power @ real @ X @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_2941_cos__one__2pi,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( one_one @ real ) )
= ( ? [X4: nat] :
( X
= ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X4 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
| ? [X4: nat] :
( X
= ( 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_2942_cos__double__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ W2 ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( sin @ A @ W2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_2943_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M6: num,N5: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M6 @ N5 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M6 ) ) @ ( unique1321980374590559556d_step @ A @ N5 @ ( unique8689654367752047608divmod @ A @ M6 @ ( bit0 @ N5 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_2944_sin__expansion__lemma,axiom,
! [X: real,M2: nat] :
( ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M2 ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_2945_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
=> ? [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_2946_cos__zero__iff,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
= ( ? [N5: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N5 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N5: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N5 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_2947_cos__expansion__lemma,axiom,
! [X: real,M2: nat] :
( ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ real @ ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M2 ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_2948_cos__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N5 ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) @ ( power_power @ real @ X @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( cos @ real @ X ) ) ).
% cos_paired
thf(fact_2949_sincos__total__2pi__le,axiom,
! [X: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
& ( X
= ( cos @ real @ T7 ) )
& ( Y
= ( sin @ real @ T7 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_2950_sincos__total__2pi,axiom,
! [X: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ~ ! [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
=> ( ( ord_less @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ( X
= ( cos @ real @ T7 ) )
=> ( Y
!= ( sin @ real @ T7 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_2951_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_2952_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_2953_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_2954_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_2955_tan__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_2956_and__int_Opsimps,axiom,
! [K2: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L ) )
=> ( ( ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K2 @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K2 @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_2957_tan__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% tan_zero
thf(fact_2958_summable__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I2: nat,F3: nat > A] :
( summable @ A
@ ^ [R: nat] : ( if @ A @ ( R = I2 ) @ ( F3 @ R ) @ ( zero_zero @ A ) ) ) ) ).
% summable_single
thf(fact_2959_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N5: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_2960_suminf__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A
@ ^ [N5: nat] : ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% suminf_zero
thf(fact_2961_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_2962_summable__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ C3 @ ( F3 @ N5 ) ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( summable @ A @ F3 ) ) ) ) ).
% summable_cmult_iff
thf(fact_2963_summable__divide__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,C3: A] :
( ( summable @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( F3 @ N5 ) @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( summable @ A @ F3 ) ) ) ) ).
% summable_divide_iff
thf(fact_2964_summable__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A6: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ A6 )
=> ( summable @ A
@ ^ [R: nat] : ( if @ A @ ( member @ nat @ R @ A6 ) @ ( F3 @ R ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite_set
thf(fact_2965_summable__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P2: nat > $o,F3: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P2 ) )
=> ( summable @ A
@ ^ [R: nat] : ( if @ A @ ( P2 @ R ) @ ( F3 @ R ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite
thf(fact_2966_tan__npi,axiom,
! [N: nat] :
( ( tan @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% tan_npi
thf(fact_2967_tan__periodic__n,axiom,
! [X: real,N: num] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ N ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_n
thf(fact_2968_tan__periodic__nat,axiom,
! [X: real,N: nat] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_nat
thf(fact_2969_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: nat > A] :
( ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N5 ) ) )
= ( F3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_2970_tan__periodic,axiom,
! [X: real] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic
thf(fact_2971_suminf__pos,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_pos
thf(fact_2972_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_2973_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ( ( suminf @ A @ F3 )
= ( zero_zero @ A ) )
= ( ! [N5: nat] :
( ( F3 @ N5 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_2974_suminf__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,C3: A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ C3 @ ( F3 @ N5 ) ) )
= ( times_times @ A @ C3 @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_mult
thf(fact_2975_suminf__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,C3: A] :
( ( summable @ A @ F3 )
=> ( ( times_times @ A @ ( suminf @ A @ F3 ) @ C3 )
= ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ C3 ) ) ) ) ) ).
% suminf_mult2
thf(fact_2976_suminf__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,G3: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G3 @ N3 ) )
=> ( ( summable @ A @ F3 )
=> ( ( summable @ A @ G3 )
=> ( ord_less_eq @ A @ ( suminf @ A @ F3 ) @ ( suminf @ A @ G3 ) ) ) ) ) ) ).
% suminf_le
thf(fact_2977_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) )
= ( ? [I: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_2978_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I2: nat] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_2979_summable__comparison__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G3: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( G3 @ N3 ) ) )
=> ( ( summable @ real @ G3 )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test
thf(fact_2980_summable__comparison__test_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G3: nat > real,N7: nat,F3: nat > A] :
( ( summable @ real @ G3 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( G3 @ N3 ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test'
thf(fact_2981_summable__const__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C3: A] :
( ( summable @ A
@ ^ [Uu3: nat] : C3 )
= ( C3
= ( zero_zero @ A ) ) ) ) ).
% summable_const_iff
thf(fact_2982_suminf__le__const,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,X: A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X )
=> ( ord_less_eq @ A @ ( suminf @ A @ F3 ) @ X ) ) ) ) ).
% suminf_le_const
thf(fact_2983_summable__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,C3: A] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ C3 ) ) ) ) ).
% summable_mult2
thf(fact_2984_summable__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,C3: A] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ C3 @ ( F3 @ N5 ) ) ) ) ) ).
% summable_mult
thf(fact_2985_summable__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A
@ ^ [N5: nat] : ( F3 @ ( suc @ N5 ) ) )
= ( summable @ A @ F3 ) ) ) ).
% summable_Suc_iff
thf(fact_2986_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N5: nat] : ( F3 @ ( suc @ N5 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F3 ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_2987_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I5: set @ nat] :
( ( summable @ A @ F3 )
=> ( ( finite_finite2 @ nat @ I5 )
=> ( ! [N3: nat] :
( ( member @ nat @ N3 @ ( uminus_uminus @ ( set @ nat ) @ I5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ I5 ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_2988_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,X: A,Z3: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ real
@ ^ [N5: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ Z3 @ N5 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_2989_summable__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N7: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N7 )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_finite
thf(fact_2990_summable__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ C3 @ ( F3 @ N5 ) ) )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_mult_D
thf(fact_2991_abs__zmult__eq__1,axiom,
! [M2: int,N: int] :
( ( ( abs_abs @ int @ ( times_times @ int @ M2 @ N ) )
= ( one_one @ int ) )
=> ( ( abs_abs @ int @ M2 )
= ( one_one @ int ) ) ) ).
% abs_zmult_eq_1
thf(fact_2992_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_2993_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_2994_abs__div,axiom,
! [Y: int,X: int] :
( ( dvd_dvd @ int @ Y @ X )
=> ( ( abs_abs @ int @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ int @ ( abs_abs @ int @ X ) @ ( abs_abs @ int @ Y ) ) ) ) ).
% abs_div
thf(fact_2995_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ Z3 @ N5 ) ) )
=> ( ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ Z3 @ N5 ) ) )
= ( plus_plus @ A @ ( F3 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ ( suc @ N5 ) ) @ ( power_power @ A @ Z3 @ N5 ) ) )
@ Z3 ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_2996_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ Z3 @ N5 ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ ( suc @ N5 ) ) @ ( power_power @ A @ Z3 @ N5 ) ) )
@ Z3 )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ Z3 @ N5 ) ) )
@ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_2997_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R3: real,F3: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ( ( summable @ A @ F3 )
=> ? [N9: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ N9 @ N4 )
=> ( ord_less @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( suminf @ A
@ ^ [I: nat] : ( F3 @ ( plus_plus @ nat @ I @ N4 ) ) ) )
@ R3 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_2998_summable__Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A3: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A3 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( summable @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A3 @ I ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ).
% summable_Cauchy_product
thf(fact_2999_Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A3: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A3 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( ( times_times @ A @ ( suminf @ A @ A3 ) @ ( suminf @ A @ B2 ) )
= ( suminf @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A3 @ I ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ) ).
% Cauchy_product
thf(fact_3000_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,N: nat,I2: nat] :
( ( summable @ A @ F3 )
=> ( ! [M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ M ) ) )
=> ( ( ord_less_eq @ nat @ N @ I2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_3001_Cauchy__product__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A3: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A3 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( sums @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A3 @ I ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I ) ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( times_times @ A @ ( suminf @ A @ A3 ) @ ( suminf @ A @ B2 ) ) ) ) ) ) ).
% Cauchy_product_sums
thf(fact_3002_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N5 ) ) ) ) ).
% summable_zero_power'
thf(fact_3003_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N5 ) ) ) ) ).
% summable_0_powser
thf(fact_3004_powser__split__head_I3_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ Z3 @ N5 ) ) )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ ( suc @ N5 ) ) @ ( power_power @ A @ Z3 @ N5 ) ) ) ) ) ).
% powser_split_head(3)
thf(fact_3005_summable__powser__split__head,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ ( suc @ N5 ) ) @ ( power_power @ A @ Z3 @ N5 ) ) )
= ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ Z3 @ N5 ) ) ) ) ) ).
% summable_powser_split_head
thf(fact_3006_summable__powser__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,M2: nat,Z3: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ ( plus_plus @ nat @ N5 @ M2 ) ) @ ( power_power @ A @ Z3 @ N5 ) ) )
= ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ Z3 @ N5 ) ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_3007_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G3: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( G3 @ N3 ) ) )
=> ( ( summable @ real @ G3 )
=> ( summable @ real
@ ^ [N5: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N5 ) ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_3008_summable__rabs__comparison__test,axiom,
! [F3: nat > real,G3: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( F3 @ N3 ) ) @ ( G3 @ N3 ) ) )
=> ( ( summable @ real @ G3 )
=> ( summable @ real
@ ^ [N5: nat] : ( abs_abs @ real @ ( F3 @ N5 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_3009_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N7: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N7 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N7 )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A @ F3 )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ N7 ) ) ) ) ) ).
% suminf_finite
thf(fact_3010_ln__mult,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( times_times @ real @ X @ Y ) )
= ( plus_plus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_3011_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,X: A,Z3: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ ( power_power @ A @ Z3 @ N5 ) ) ) ) ) ) ).
% powser_inside
thf(fact_3012_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,X: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X )
=> ( summable @ A @ F3 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_3013_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A3: nat > A,B6: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A3 @ ( set_ord_atMost @ nat @ N3 ) ) @ B6 )
=> ( summable @ A @ A3 ) ) ) ) ).
% bounded_imp_summable
thf(fact_3014_zdvd__mult__cancel1,axiom,
! [M2: int,N: int] :
( ( M2
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ ( times_times @ int @ M2 @ N ) @ M2 )
= ( ( abs_abs @ int @ N )
= ( one_one @ int ) ) ) ) ).
% zdvd_mult_cancel1
thf(fact_3015_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_3016_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_realpow
thf(fact_3017_summable__partial__sum__bound,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,E3: real] :
( ( summable @ A @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ~ ! [N9: nat] :
~ ! [M3: nat] :
( ( ord_less_eq @ nat @ N9 @ M3 )
=> ! [N4: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ M3 @ N4 ) ) ) @ E3 ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_3018_summable__power__series,axiom,
! [F3: nat > real,Z3: 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 ) @ Z3 )
=> ( ( ord_less @ real @ Z3 @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I: nat] : ( times_times @ real @ ( F3 @ I ) @ ( power_power @ real @ Z3 @ I ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_3019_Abel__lemma,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R3: real,R0: real,A3: nat > A,M5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R3 )
=> ( ( ord_less @ real @ R3 @ R0 )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A3 @ N3 ) ) @ ( power_power @ real @ R0 @ N3 ) ) @ M5 )
=> ( summable @ real
@ ^ [N5: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A3 @ N5 ) ) @ ( power_power @ real @ R3 @ N5 ) ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_3020_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_3021_incr__lemma,axiom,
! [D3: int,Z3: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ Z3 @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z3 ) ) @ ( one_one @ int ) ) @ D3 ) ) ) ) ).
% incr_lemma
thf(fact_3022_decr__lemma,axiom,
! [D3: int,X: int,Z3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z3 ) ) @ ( one_one @ int ) ) @ D3 ) ) @ Z3 ) ) ).
% decr_lemma
thf(fact_3023_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C3: real,N7: nat,F3: nat > A] :
( ( ord_less @ real @ C3 @ ( one_one @ real ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ ( suc @ N3 ) ) ) @ ( times_times @ real @ C3 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_ratio_test
thf(fact_3024_ln__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ( ln_ln @ real @ X )
= ( suminf @ real
@ ^ [N5: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N5 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ ( one_one @ real ) ) @ ( suc @ N5 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_3025_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_3026_add__tan__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) )
= ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_3027_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_3028_tan__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( plus_plus @ A @ X @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( plus_plus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_3029_tan__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( minus_minus @ A @ X @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( minus_minus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_3030_lemma__tan__add1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) )
= ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_3031_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_3032_ln__one__minus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( uminus_uminus @ real @ X ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X ) ) ) ) ) ).
% ln_one_minus_pos_lower_bound
thf(fact_3033_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound
thf(fact_3034_and__int_Opelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa2 ) )
=> ~ ( ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa2 ) ) ) ) ) ).
% and_int.pelims
thf(fact_3035_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arctan @ X )
= ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_3036_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C3: nat > A,X: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N5 ) @ ( C3 @ N5 ) ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_3037_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M6 ) @ ( semiring_char_0_fact @ real @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_3038_arctan__double,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X ) )
= ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_3039_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X7: nat > A] :
( ! [M6: nat,N5: nat] :
( ( ord_less_eq @ nat @ M6 @ N5 )
=> ( ord_less_eq @ A @ ( X7 @ M6 ) @ ( X7 @ N5 ) ) )
| ! [M6: nat,N5: nat] :
( ( ord_less_eq @ nat @ M6 @ N5 )
=> ( ord_less_eq @ A @ ( X7 @ N5 ) @ ( X7 @ M6 ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_3040_exp__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% exp_zero
thf(fact_3041_exp__not__eq__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( exp @ A @ X )
!= ( zero_zero @ A ) ) ) ).
% exp_not_eq_zero
thf(fact_3042_exp__times__arg__commute,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [A6: A] :
( ( times_times @ A @ ( exp @ A @ A6 ) @ A6 )
= ( times_times @ A @ A6 @ ( exp @ A @ A6 ) ) ) ) ).
% exp_times_arg_commute
thf(fact_3043_mult__exp__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ Y ) )
= ( exp @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% mult_exp_exp
thf(fact_3044_exp__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A,Y: A] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( exp @ A @ ( plus_plus @ A @ X @ Y ) )
= ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ Y ) ) ) ) ) ).
% exp_add_commuting
thf(fact_3045_exp__minus__inverse,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) )
= ( one_one @ A ) ) ) ).
% exp_minus_inverse
thf(fact_3046_exp__of__nat__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X: A] :
( ( exp @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ X ) )
= ( power_power @ A @ ( exp @ A @ X ) @ N ) ) ) ).
% exp_of_nat_mult
thf(fact_3047_exp__of__nat2__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,N: nat] :
( ( exp @ A @ ( times_times @ A @ X @ ( semiring_1_of_nat @ A @ N ) ) )
= ( power_power @ A @ ( exp @ A @ X ) @ N ) ) ) ).
% exp_of_nat2_mult
thf(fact_3048_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C4: nat > A,N5: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N5 ) ) @ ( C4 @ ( suc @ N5 ) ) ) ) ) ) ).
% diffs_def
thf(fact_3049_termdiff__converges__all,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,X: A] :
( ! [X3: A] :
( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ X3 @ N5 ) ) )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) ) ) ) ).
% termdiff_converges_all
thf(fact_3050_exp__divide__power__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X @ ( semiring_1_of_nat @ A @ N ) ) ) @ N )
= ( exp @ A @ X ) ) ) ) ).
% exp_divide_power_eq
thf(fact_3051_exp__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z3: A] :
( ( exp @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z3 ) )
= ( power_power @ A @ ( exp @ A @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% exp_double
thf(fact_3052_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,K5: real,C3: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ K5 )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K5 )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ X3 @ N5 ) ) ) )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_3053_real__exp__bound__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) ) ) ) ) ).
% real_exp_bound_lemma
thf(fact_3054_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ X ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_3055_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ ( uminus_uminus @ real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_3056_arctan__add,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( plus_plus @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X @ Y ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X @ Y ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_3057_exp__bound__lemma,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z3: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ Z3 ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( real_V7770717601297561774m_norm @ A @ Z3 ) ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_3058_Maclaurin__exp__le,axiom,
! [X: real,N: nat] :
? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M6 ) @ ( semiring_char_0_fact @ real @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_3059_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_3060_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_3061_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% mono_SucI1
thf(fact_3062_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% mono_SucI2
thf(fact_3063_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X7: nat > A] :
( ! [N5: nat] : ( ord_less_eq @ A @ ( X7 @ N5 ) @ ( X7 @ ( suc @ N5 ) ) )
| ! [N5: nat] : ( ord_less_eq @ A @ ( X7 @ ( suc @ N5 ) ) @ ( X7 @ N5 ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_3064_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ M ) @ ( X8 @ N3 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% monoI1
thf(fact_3065_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ M ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% monoI2
thf(fact_3066_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_3067_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N5 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) ) @ ( power_power @ A @ X @ N5 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N5 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ).
% sin_x_sin_y
thf(fact_3068_Maclaurin__sin__bound,axiom,
! [X: real,N: nat] :
( ord_less_eq @ real
@ ( abs_abs @ real
@ ( minus_minus @ real @ ( sin @ real @ X )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( sin_coeff @ M6 ) @ ( power_power @ real @ X @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) )
@ ( times_times @ real @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( abs_abs @ real @ X ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_3069_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: 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 @ N5 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X @ N5 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N5 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_3070_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_3071_scaleR__cancel__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A,B2: real] :
( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
= ( ( A3 = B2 )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_cancel_right
thf(fact_3072_scaleR__zero__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_right
thf(fact_3073_inverse__nonzero__iff__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_3074_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_3075_mult__scaleR__left,axiom,
! [A: $tType] :
( ( real_V6157519004096292374lgebra @ A )
=> ! [A3: real,X: A,Y: A] :
( ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ Y )
= ( real_V8093663219630862766scaleR @ A @ A3 @ ( times_times @ A @ X @ Y ) ) ) ) ).
% mult_scaleR_left
thf(fact_3076_mult__scaleR__right,axiom,
! [A: $tType] :
( ( real_V6157519004096292374lgebra @ A )
=> ! [X: A,A3: real,Y: A] :
( ( times_times @ A @ X @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y ) )
= ( real_V8093663219630862766scaleR @ A @ A3 @ ( times_times @ A @ X @ Y ) ) ) ) ).
% mult_scaleR_right
thf(fact_3077_inverse__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% inverse_mult_distrib
thf(fact_3078_scaleR__scaleR,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,B2: real,X: A] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
= ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ A3 @ B2 ) @ X ) ) ) ).
% scaleR_scaleR
thf(fact_3079_tanh__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% tanh_0
thf(fact_3080_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_3081_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_3082_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_3083_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_3084_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_3085_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_3086_scaleR__zero__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( zero_zero @ real ) @ X )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_left
thf(fact_3087_scaleR__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ real ) )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_eq_0_iff
thf(fact_3088_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z3: int] :
( ( ( ring_1_of_int @ A @ Z3 )
= ( zero_zero @ A ) )
= ( Z3
= ( zero_zero @ int ) ) ) ) ).
% of_int_eq_0_iff
thf(fact_3089_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z3: int] :
( ( ( zero_zero @ A )
= ( ring_1_of_int @ A @ Z3 ) )
= ( Z3
= ( zero_zero @ int ) ) ) ) ).
% of_int_0_eq_iff
thf(fact_3090_of__int__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
= ( zero_zero @ A ) ) ) ).
% of_int_0
thf(fact_3091_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less_eq @ int @ W2 @ Z3 ) ) ) ).
% of_int_le_iff
thf(fact_3092_of__int__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z3: int] :
( ( ring_1_of_int @ A @ ( times_times @ int @ W2 @ Z3 ) )
= ( times_times @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_mult
thf(fact_3093_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_3094_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_3095_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_3096_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_3097_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_3098_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num] :
( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_3099_norm__scaleR,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: real,X: A] :
( ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) )
= ( times_times @ real @ ( abs_abs @ real @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ) ).
% norm_scaleR
thf(fact_3100_pred__numeral__simps_I2_J,axiom,
! [K2: num] :
( ( pred_numeral @ ( bit0 @ K2 ) )
= ( numeral_numeral @ nat @ ( bitM @ K2 ) ) ) ).
% pred_numeral_simps(2)
thf(fact_3101_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_3102_of__int__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) ) ) ) ).
% of_int_le_0_iff
thf(fact_3103_of__int__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% of_int_0_le_iff
thf(fact_3104_of__int__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( zero_zero @ A ) )
= ( ord_less @ int @ Z3 @ ( zero_zero @ int ) ) ) ) ).
% of_int_less_0_iff
thf(fact_3105_of__int__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% of_int_0_less_iff
thf(fact_3106_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z3: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z3 ) ) ) ).
% of_int_numeral_le_iff
thf(fact_3107_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int,N: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ int @ Z3 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_3108_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% of_int_1_le_iff
thf(fact_3109_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z3 @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_3110_scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U: num,W2: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ U ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W2 ) ) @ A3 ) ) ) ).
% scaleR_times
thf(fact_3111_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_3112_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_3113_sin__npi__int,axiom,
! [N: int] :
( ( sin @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_npi_int
thf(fact_3114_tan__periodic__int,axiom,
! [X: real,I2: int] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( ring_1_of_int @ real @ I2 ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_int
thf(fact_3115_inverse__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [V3: num,W2: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ V3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ W2 ) @ ( numeral_numeral @ real @ V3 ) ) @ A3 ) ) ) ).
% inverse_scaleR_times
thf(fact_3116_fraction__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U: num,V3: num,W2: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ V3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W2 ) ) @ ( numeral_numeral @ real @ V3 ) ) @ A3 ) ) ) ).
% fraction_scaleR_times
thf(fact_3117_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_3118_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_3119_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_3120_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_3121_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_3122_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_3123_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_3124_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Y: A,X: A] :
( ( ( times_times @ A @ Y @ X )
= ( times_times @ A @ X @ Y ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ Y ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ Y ) ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_3125_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ).
% ex_le_of_int
thf(fact_3126_nonzero__norm__inverse,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ A3 ) )
= ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_3127_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa2: int,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa2 ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa2 ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3128_nonzero__inverse__scaleR__distrib,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A3: real,X: A] :
( ( A3
!= ( zero_zero @ real ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) )
= ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ A3 ) @ ( inverse_inverse @ A @ X ) ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_3129_field__class_Ofield__inverse__zero,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% field_class.field_inverse_zero
thf(fact_3130_inverse__zero__imp__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ).
% inverse_zero_imp_zero
thf(fact_3131_nonzero__inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( inverse_inverse @ A @ B2 ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( A3 = B2 ) ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_3132_nonzero__inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A3 ) )
= A3 ) ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_3133_nonzero__imp__inverse__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
!= ( zero_zero @ A ) ) ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_3134_scaleR__right__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,A3: real,B2: real] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
=> ( A3 = B2 ) ) ) ) ).
% scaleR_right_imp_eq
thf(fact_3135_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: int,Y: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% mult_of_int_commute
thf(fact_3136_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ).
% ex_of_int_less
thf(fact_3137_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ).
% ex_less_of_int
thf(fact_3138_real__scaleR__def,axiom,
( ( real_V8093663219630862766scaleR @ real )
= ( times_times @ real ) ) ).
% real_scaleR_def
thf(fact_3139_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_3140_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_3141_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_3142_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_3143_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_3144_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_3145_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_3146_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_3147_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_3148_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_3149_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_3150_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_3151_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_3152_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_3153_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_3154_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_3155_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_3156_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_3157_nonzero__inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_3158_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= B2 ) ) ) ).
% inverse_unique
thf(fact_3159_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A7: A,B5: A] : ( times_times @ A @ A7 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_3160_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A7: A,B5: A] : ( times_times @ A @ A7 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% divide_inverse
thf(fact_3161_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A7: A,B5: A] : ( times_times @ A @ ( inverse_inverse @ A @ B5 ) @ A7 ) ) ) ) ).
% divide_inverse_commute
thf(fact_3162_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( inverse_inverse @ A @ X ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M2 ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_3163_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) @ ( power_power @ A @ X @ M2 ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_3164_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa2: nat,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa2 ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa2 ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_3165_nonzero__abs__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( inverse_inverse @ A @ A3 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A3 ) ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_3166_divide__real__def,axiom,
( ( divide_divide @ real )
= ( ^ [X4: real,Y4: real] : ( times_times @ real @ X4 @ ( inverse_inverse @ real @ Y4 ) ) ) ) ).
% divide_real_def
thf(fact_3167_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_3168_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_3169_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_3170_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_3171_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_le_1_iff
thf(fact_3172_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_3173_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_less_inverse
thf(fact_3174_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_3175_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_3176_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_3177_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ B2 @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_3178_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_3179_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_3180_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: real,X: A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_3181_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_3182_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_3183_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_3184_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,Y: A,A3: real] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_3185_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_3186_eq__vector__fraction__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,U: real,V3: real,A3: A] :
( ( X
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V3 ) @ A3 ) )
= ( ( ( V3
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V3
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ V3 @ X )
= ( real_V8093663219630862766scaleR @ A @ U @ A3 ) ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_3187_vector__fraction__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,V3: real,A3: A,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V3 ) @ A3 )
= X )
= ( ( ( V3
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V3
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ U @ A3 )
= ( real_V8093663219630862766scaleR @ A @ V3 @ X ) ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_3188_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_3189_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_3190_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_3191_exp__series__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I ) ) @ ( power_power @ A @ X @ I ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ I ) ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ I ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_3192_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
@ ^ [N5: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N5 ) ) ) @ ( power_power @ A @ X4 @ ( suc @ N5 ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_3193_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_3194_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_3195_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_3196_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_less_1_iff
thf(fact_3197_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_3198_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_3199_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_3200_inverse__diff__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ A3 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_3201_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_3202_of__int__nonneg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_nonneg
thf(fact_3203_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_leD
thf(fact_3204_of__int__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_pos
thf(fact_3205_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_3206_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_3207_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_3208_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_3209_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_3210_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_3211_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_3212_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_3213_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_3214_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: real,X: A,Y: A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_3215_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,A3: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ real @ A3 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ X ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_3216_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [X3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) ) )
& ! [Y6: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y6 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y6 @ ( one_one @ int ) ) ) ) )
=> ( Y6 = X3 ) ) ) ) ).
% floor_exists1
thf(fact_3217_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_3218_forall__pos__mono__1,axiom,
! [P2: real > $o,E3: real] :
( ! [D2: real,E2: real] :
( ( ord_less @ real @ D2 @ E2 )
=> ( ( P2 @ D2 )
=> ( P2 @ E2 ) ) )
=> ( ! [N3: nat] : ( P2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( P2 @ E3 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_3219_forall__pos__mono,axiom,
! [P2: real > $o,E3: real] :
( ! [D2: real,E2: real] :
( ( ord_less @ real @ D2 @ E2 )
=> ( ( P2 @ D2 )
=> ( P2 @ E2 ) ) )
=> ( ! [N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
=> ( P2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( P2 @ E3 ) ) ) ) ).
% forall_pos_mono
thf(fact_3220_real__arch__inverse,axiom,
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
= ( ? [N5: nat] :
( ( N5
!= ( zero_zero @ nat ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N5 ) ) )
& ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N5 ) ) @ E3 ) ) ) ) ).
% real_arch_inverse
thf(fact_3221_sin__zero__iff__int2,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I: int] :
( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I ) @ pi ) ) ) ) ).
% sin_zero_iff_int2
thf(fact_3222_summable__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N5 ) ) @ ( power_power @ A @ X @ N5 ) ) ) ) ).
% summable_exp
thf(fact_3223_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_3224_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N3 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_3225_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N @ M2 ) )
= ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M2 ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_3226_cos__one__2pi__int,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( one_one @ real ) )
= ( ? [X4: int] :
( X
= ( times_times @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ X4 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_3227_tan__sec,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tan_sec
thf(fact_3228_cos__zero__iff__int,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I: int] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_3229_sin__zero__iff__int,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_3230_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P5: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P5 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P5 @ N5 ) ) ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ A @ X @ N5 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P5 @ N5 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P5 ) )
@ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ).
% cos_x_cos_y
thf(fact_3231_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_3232_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X )
= Y ) ) ) ) ).
% round_unique
thf(fact_3233_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X )
= N ) ) ) ).
% round_unique'
thf(fact_3234_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ X ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_3235_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N5: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N5 ) ) @ ( power_power @ A @ X @ N5 ) ) )
@ ( sinh @ A @ X ) ) ) ).
% sinh_converges
thf(fact_3236_sinh__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sinh_0
thf(fact_3237_round__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int] :
( ( archimedean_round @ A @ ( ring_1_of_int @ A @ N ) )
= N ) ) ).
% round_of_int
thf(fact_3238_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_3239_round__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ int @ N ) ) ) ).
% round_numeral
thf(fact_3240_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_3241_round__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archimedean_round @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% round_of_nat
thf(fact_3242_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_3243_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% round_mono
thf(fact_3244_round__diff__minimal,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: A,M2: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z3 @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z3 ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z3 @ ( ring_1_of_int @ A @ M2 ) ) ) ) ) ).
% round_diff_minimal
thf(fact_3245_sinh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sinh @ A @ X )
= ( zero_zero @ A ) )
= ( member @ A @ ( exp @ A @ X ) @ ( insert @ A @ ( one_one @ A ) @ ( insert @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% sinh_zero_iff
thf(fact_3246_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_3247_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_ge
thf(fact_3248_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_gt
thf(fact_3249_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N5: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N5 ) ) @ ( power_power @ A @ X @ N5 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X ) ) ) ).
% cosh_converges
thf(fact_3250_complex__unimodular__polar,axiom,
! [Z3: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z3 )
= ( one_one @ real ) )
=> ~ ! [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
=> ( ( ord_less @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( Z3
!= ( complex2 @ ( cos @ real @ T7 ) @ ( sin @ real @ T7 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_3251_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_3252_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_3253_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log2 @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq1
thf(fact_3254_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_3255_cosh__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% cosh_0
thf(fact_3256_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_3257_complex__scaleR,axiom,
! [R3: real,A3: real,B2: real] :
( ( real_V8093663219630862766scaleR @ complex @ R3 @ ( complex2 @ A3 @ B2 ) )
= ( complex2 @ ( times_times @ real @ R3 @ A3 ) @ ( times_times @ real @ R3 @ B2 ) ) ) ).
% complex_scaleR
thf(fact_3258_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F3: A > B > C,G3: A > B > C,P: product_prod @ A @ B] :
( ! [X3: A,Y3: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y3 )
= Q2 )
=> ( ( F3 @ X3 @ Y3 )
= ( G3 @ X3 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F3 @ P )
= ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% split_cong
thf(fact_3259_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_3260_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
@ ^ [X16: A,X24: B] : ( H @ ( F3 @ X16 @ X24 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_3261_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C,G3: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y3: B] :
( ( F3 @ X3 @ Y3 )
= ( G3 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F3 )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_3262_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X4: A,Y4: B] : ( F3 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_3263_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P2: B > C > A,Z3: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P2 @ Z3 ) )
=> ~ ! [X3: B,Y3: C] :
( ( Z3
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_3264_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc5280177257484947105e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_3265_complex__mult,axiom,
! [A3: real,B2: real,C3: real,D3: real] :
( ( times_times @ complex @ ( complex2 @ A3 @ B2 ) @ ( complex2 @ C3 @ D3 ) )
= ( complex2 @ ( minus_minus @ real @ ( times_times @ real @ A3 @ C3 ) @ ( times_times @ real @ B2 @ D3 ) ) @ ( plus_plus @ real @ ( times_times @ real @ A3 @ D3 ) @ ( times_times @ real @ B2 @ C3 ) ) ) ) ).
% complex_mult
thf(fact_3266_sinh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( sinh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( cosh @ A @ X ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% sinh_add
thf(fact_3267_cosh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cosh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cosh @ A @ X ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( sinh @ A @ X ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% cosh_add
thf(fact_3268_sinh__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( sinh @ A @ ( minus_minus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( times_times @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( cosh @ A @ X ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% sinh_diff
thf(fact_3269_cosh__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cosh @ A @ ( minus_minus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cosh @ A @ X ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( sinh @ A @ X ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% cosh_diff
thf(fact_3270_log__mult,axiom,
! [A3: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log2 @ A3 @ ( times_times @ real @ X @ Y ) )
= ( plus_plus @ real @ ( log2 @ A3 @ X ) @ ( log2 @ A3 @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_3271_log__nat__power,axiom,
! [X: real,B2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log2 @ B2 @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log2 @ B2 @ X ) ) ) ) ).
% log_nat_power
thf(fact_3272_sinh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( sinh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sinh @ A @ X ) ) @ ( cosh @ A @ X ) ) ) ) ).
% sinh_double
thf(fact_3273_log__of__power__less,axiom,
! [M2: nat,B2: real,N: nat] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ real @ ( log2 @ B2 @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_3274_log__eq__div__ln__mult__log,axiom,
! [A3: real,B2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log2 @ A3 @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( ln_ln @ real @ A3 ) ) @ ( log2 @ B2 @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_3275_log__of__power__le,axiom,
! [M2: nat,B2: real,N: nat] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less_eq @ real @ ( log2 @ B2 @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_3276_tanh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cosh @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cosh @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( tanh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y ) ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_3277_le__log2__of__power,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M2 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% le_log2_of_power
thf(fact_3278_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat,R: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_3279_log2__of__power__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ real @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_less
thf(fact_3280_cosh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cosh @ A @ X )
= ( zero_zero @ A ) )
= ( ( power_power @ A @ ( exp @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% cosh_zero_iff
thf(fact_3281_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q5: int,R: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ R ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_3282_cosh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( cosh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( plus_plus @ A @ ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_double
thf(fact_3283_log2__of__power__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less_eq @ real @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_le
thf(fact_3284_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log2 @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq2
thf(fact_3285_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_3286_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 @ ( log2 @ ( 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_3287_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 @ ( log2 @ ( 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_3288_ceiling__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archimedean_ceiling @ real @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ real @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% ceiling_log2_div2
thf(fact_3289_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M6: nat,N5: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N5
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M6 @ N5 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M6 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M6 @ N5 ) @ N5 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_3290_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 @ ( log2 @ ( 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_3291_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_3292_case__prodI2,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,C3: A > B > $o] :
( ! [A5: A,B4: B] :
( ( P
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( C3 @ A5 @ B4 ) )
=> ( product_case_prod @ A @ B @ $o @ C3 @ P ) ) ).
% case_prodI2
thf(fact_3293_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z3: A,C3: B > C > ( set @ A ),A3: B,B2: C] :
( ( member @ A @ Z3 @ ( C3 @ A3 @ B2 ) )
=> ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A3 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_3294_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P: product_prod @ A @ B,Z3: C,C3: A > B > ( set @ C )] :
( ! [A5: A,B4: B] :
( ( P
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( member @ C @ Z3 @ ( C3 @ A5 @ B4 ) ) )
=> ( member @ C @ Z3 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_3295_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: product_prod @ A @ B,C3: A > B > C > $o,X: C] :
( ! [A5: A,B4: B] :
( ( ( product_Pair @ A @ B @ A5 @ B4 )
= P )
=> ( C3 @ A5 @ B4 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P @ X ) ) ).
% case_prodI2'
thf(fact_3296_floor__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int] :
( ( archim6421214686448440834_floor @ A @ ( ring_1_of_int @ A @ Z3 ) )
= Z3 ) ) ).
% floor_of_int
thf(fact_3297_ceiling__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int] :
( ( archimedean_ceiling @ A @ ( ring_1_of_int @ A @ Z3 ) )
= Z3 ) ) ).
% ceiling_of_int
thf(fact_3298_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_3299_floor__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num] :
( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ int @ V3 ) ) ) ).
% floor_numeral
thf(fact_3300_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_3301_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_3302_ceiling__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num] :
( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ int @ V3 ) ) ) ).
% ceiling_numeral
thf(fact_3303_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_3304_floor__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archim6421214686448440834_floor @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% floor_of_nat
thf(fact_3305_ceiling__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% ceiling_of_nat
thf(fact_3306_floor__uminus__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z3 ) ) )
= ( uminus_uminus @ int @ Z3 ) ) ) ).
% floor_uminus_of_int
thf(fact_3307_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 ) ) ) ).
% ceiling_add_of_int
thf(fact_3308_floor__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 ) ) ) ).
% floor_diff_of_int
thf(fact_3309_ceiling__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 ) ) ) ).
% ceiling_diff_of_int
thf(fact_3310_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_le_floor
thf(fact_3311_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_3312_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V3 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V3 ) @ X ) ) ) ).
% numeral_le_floor
thf(fact_3313_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% zero_less_floor
thf(fact_3314_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_3315_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_3316_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V3 ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ V3 ) ) ) ) ).
% floor_less_numeral
thf(fact_3317_zero__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_less_ceiling
thf(fact_3318_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_le_floor
thf(fact_3319_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V3 ) )
= ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V3 ) ) ) ) ).
% ceiling_le_numeral
thf(fact_3320_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_3321_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_3322_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V3 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V3 ) @ X ) ) ) ).
% numeral_less_ceiling
thf(fact_3323_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% one_le_ceiling
thf(fact_3324_floor__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) ) ) ).
% floor_neg_numeral
thf(fact_3325_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_3326_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_less_ceiling
thf(fact_3327_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( numeral_numeral @ A @ V3 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V3 ) ) ) ) ).
% ceiling_add_numeral
thf(fact_3328_floor__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V3 ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V3 ) ) ) ) ).
% floor_diff_numeral
thf(fact_3329_ceiling__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) ) ) ).
% ceiling_neg_numeral
thf(fact_3330_ceiling__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_add_one
thf(fact_3331_floor__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% floor_diff_one
thf(fact_3332_ceiling__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V3 ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V3 ) ) ) ) ).
% ceiling_diff_numeral
thf(fact_3333_ceiling__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_diff_one
thf(fact_3334_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_3335_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X ) ) ) ).
% zero_le_ceiling
thf(fact_3336_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V3 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_less_floor
thf(fact_3337_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V3 ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V3 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_3338_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% one_less_floor
thf(fact_3339_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_3340_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V3 ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V3 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_3341_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V3 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_le_ceiling
thf(fact_3342_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ X ) ) ) ).
% neg_numeral_le_floor
thf(fact_3343_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) )
= ( ord_less @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_3344_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_3345_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ X ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_3346_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_less_floor
thf(fact_3347_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_3348_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V3: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_3349_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V3: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V3 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_3350_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z3: A,C3: B > C > ( set @ A ),P: product_prod @ B @ C] :
( ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P ) )
=> ~ ! [X3: B,Y3: C] :
( ( P
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( member @ A @ Z3 @ ( C3 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_3351_floor__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% floor_le_ceiling
thf(fact_3352_ceiling__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X4: A] : ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X4 ) ) ) ) ) ) ).
% ceiling_def
thf(fact_3353_floor__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ int @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% floor_minus
thf(fact_3354_ceiling__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ).
% ceiling_minus
thf(fact_3355_divide__complex__def,axiom,
( ( divide_divide @ complex )
= ( ^ [X4: complex,Y4: complex] : ( times_times @ complex @ X4 @ ( inverse_inverse @ complex @ Y4 ) ) ) ) ).
% divide_complex_def
thf(fact_3356_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_3357_case__prodE,axiom,
! [A: $tType,B: $tType,C3: A > B > $o,P: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C3 @ P )
=> ~ ! [X3: A,Y3: B] :
( ( P
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C3 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_3358_ceiling__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X4: A] :
( if @ int
@ ( X4
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X4 ) ) )
@ ( archim6421214686448440834_floor @ A @ X4 )
@ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X4 ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_altdef
thf(fact_3359_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) @ ( one_one @ int ) ) ) ).
% ceiling_diff_floor_le_1
thf(fact_3360_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R2: A > B > C > $o,A3: A,B2: B,C3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R2 @ ( product_Pair @ A @ B @ A3 @ B2 ) @ C3 )
=> ( R2 @ A3 @ B2 @ C3 ) ) ).
% case_prodD'
thf(fact_3361_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P: product_prod @ A @ B,Z3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P @ Z3 )
=> ~ ! [X3: A,Y3: B] :
( ( P
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C3 @ X3 @ Y3 @ Z3 ) ) ) ).
% case_prodE'
thf(fact_3362_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ).
% floor_mono
thf(fact_3363_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X ) ) ).
% of_int_floor_le
thf(fact_3364_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% floor_less_cancel
thf(fact_3365_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% ceiling_mono
thf(fact_3366_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% le_of_int_ceiling
thf(fact_3367_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% ceiling_less_cancel
thf(fact_3368_floor__le__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_round @ A @ X ) ) ) ).
% floor_le_round
thf(fact_3369_ceiling__ge__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% ceiling_ge_round
thf(fact_3370_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
@ ^ [I: nat,J: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ J ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( G3 @ I @ ( minus_minus @ nat @ K3 @ I ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_3371_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
@ ^ [I: nat,J: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ J ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( G3 @ I @ ( minus_minus @ nat @ K3 @ I ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_3372_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less_eq @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ).
% le_floor_iff
thf(fact_3373_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
= ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% floor_less_iff
thf(fact_3374_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% le_floor_add
thf(fact_3375_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ) ).
% floor_add_int
thf(fact_3376_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( plus_plus @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ) ).
% int_add_floor
thf(fact_3377_floor__divide__of__int__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [K2: int,L: int] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K2 ) @ ( ring_1_of_int @ A @ L ) ) )
= ( divide_divide @ int @ K2 @ L ) ) ) ).
% floor_divide_of_int_eq
thf(fact_3378_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A3 ) ) ) ).
% ceiling_le
thf(fact_3379_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% ceiling_le_iff
thf(fact_3380_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less @ int @ Z3 @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ).
% less_ceiling_iff
thf(fact_3381_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ).
% ceiling_add_le
thf(fact_3382_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
@ ^ [I: nat,J: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( G3 @ I @ ( minus_minus @ nat @ K3 @ I ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_3383_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
@ ^ [I: nat,J: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( G3 @ I @ ( minus_minus @ nat @ K3 @ I ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_3384_one__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_add_floor
thf(fact_3385_floor__divide__of__nat__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [M2: nat,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M2 @ N ) ) ) ) ).
% floor_divide_of_nat_eq
thf(fact_3386_of__int__ceiling__le__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R3: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R3 ) ) @ ( plus_plus @ A @ R3 @ ( one_one @ A ) ) ) ) ).
% of_int_ceiling_le_add_one
thf(fact_3387_of__int__ceiling__diff__one__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R3 ) ) @ ( one_one @ A ) ) @ R3 ) ) ).
% of_int_ceiling_diff_one_le
thf(fact_3388_prod__encode__def,axiom,
( nat_prod_encode
= ( product_case_prod @ nat @ nat @ nat
@ ^ [M6: nat,N5: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M6 @ N5 ) ) @ M6 ) ) ) ).
% prod_encode_def
thf(fact_3389_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P2: int > $o,T2: A] :
( ( P2 @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I ) @ ( one_one @ A ) ) ) )
=> ( P2 @ I ) ) ) ) ) ).
% floor_split
thf(fact_3390_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A3 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_3391_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
=> ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X )
= Z3 ) ) ) ) ).
% floor_unique
thf(fact_3392_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_3393_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% less_floor_iff
thf(fact_3394_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_3395_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_3396_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P2: int > $o,T2: A] :
( ( P2 @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I ) ) )
=> ( P2 @ I ) ) ) ) ) ).
% ceiling_split
thf(fact_3397_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archimedean_ceiling @ A @ X )
= A3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_3398_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z3 ) ) ) ) ).
% ceiling_unique
thf(fact_3399_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% ceiling_correct
thf(fact_3400_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_3401_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_3402_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less_eq @ int @ Z3 @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_3403_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P @ Q2 ) ) ) @ Q2 ) @ P ) ) ) ).
% floor_divide_lower
thf(fact_3404_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q5: int,R: int] :
( plus_plus @ int @ Q5
@ ( zero_neq_one_of_bool @ int
@ ( R
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_3405_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [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 @ R ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_3406_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less_eq @ A @ P @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P @ Q2 ) ) ) @ Q2 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_3407_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less @ A @ P @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) ) ) ) ).
% floor_divide_upper
thf(fact_3408_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_3409_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) @ P ) ) ) ).
% ceiling_divide_lower
thf(fact_3410_ceiling__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ N ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N ) @ ( one_one @ A ) ) )
=> ( ( archimedean_ceiling @ A @ X )
= ( plus_plus @ int @ N @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_eq
thf(fact_3411_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M6: nat,N5: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M6 @ N5 ) @ ( modulo_modulo @ nat @ M6 @ N5 ) ) ) ) ).
% divmod_nat_def
thf(fact_3412_floor__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archim6421214686448440834_floor @ real @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ real @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% floor_log2_div2
thf(fact_3413_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 @ ( log2 @ ( 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_3414_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N5: nat] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M6: nat,Q5: nat] :
( if @ A
@ ( Q5
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M6 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M6 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_3415_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_3416_cot__periodic,axiom,
! [X: real] :
( ( cot @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cot @ real @ X ) ) ).
% cot_periodic
thf(fact_3417_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_3418_modulo__int__unfold,axiom,
! [L: int,K2: int,N: nat,M2: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K2 )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( semiring_1_of_nat @ int @ M2 ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K2 )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ) )
& ( ( ( sgn_sgn @ int @ K2 )
!= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L )
@ ( minus_minus @ int
@ ( semiring_1_of_nat @ int
@ ( times_times @ nat @ N
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M2 ) ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_3419_sgn__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A3 ) )
= ( sgn_sgn @ A @ A3 ) ) ) ).
% sgn_sgn
thf(fact_3420_frac__frac,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_frac @ A @ ( archimedean_frac @ A @ X ) )
= ( archimedean_frac @ A @ X ) ) ) ).
% frac_frac
thf(fact_3421_split__part,axiom,
! [B: $tType,A: $tType,P2: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] :
( P2
& ( Q @ A7 @ B5 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P2
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_3422_sgn__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_3423_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_3424_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_3425_idom__abs__sgn__class_Osgn__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ A3 ) ) ) ) ).
% idom_abs_sgn_class.sgn_minus
thf(fact_3426_cot__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cot @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% cot_zero
thf(fact_3427_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_greater
thf(fact_3428_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_3429_divide__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ A3 @ ( sgn_sgn @ A @ B2 ) )
= ( times_times @ A @ A3 @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% divide_sgn
thf(fact_3430_frac__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int] :
( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z3 ) )
= ( zero_zero @ A ) ) ) ).
% frac_of_int
thf(fact_3431_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_3432_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_3433_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_3434_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( abs_abs @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_3435_sgn__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_abs
thf(fact_3436_dvd__mult__sgn__iff,axiom,
! [L: int,K2: int,R3: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ K2 @ ( sgn_sgn @ int @ R3 ) ) )
= ( ( dvd_dvd @ int @ L @ K2 )
| ( R3
= ( zero_zero @ int ) ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_3437_dvd__sgn__mult__iff,axiom,
! [L: int,R3: int,K2: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R3 ) @ K2 ) )
= ( ( dvd_dvd @ int @ L @ K2 )
| ( R3
= ( zero_zero @ int ) ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_3438_mult__sgn__dvd__iff,axiom,
! [L: int,R3: int,K2: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R3 ) ) @ K2 )
= ( ( dvd_dvd @ int @ L @ K2 )
& ( ( R3
= ( zero_zero @ int ) )
=> ( K2
= ( zero_zero @ int ) ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_3439_sgn__mult__dvd__iff,axiom,
! [R3: int,L: int,K2: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R3 ) @ L ) @ K2 )
= ( ( dvd_dvd @ int @ L @ K2 )
& ( ( R3
= ( zero_zero @ int ) )
=> ( K2
= ( zero_zero @ int ) ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_3440_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_3441_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_3442_cot__npi,axiom,
! [N: nat] :
( ( cot @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% cot_npi
thf(fact_3443_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A6: A > B > $o,B6: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A6 @ B6 )
=> ( 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 @ B6 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_3444_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_3445_Real__Vector__Spaces_Osgn__mult,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,Y: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ X @ Y ) )
= ( times_times @ A @ ( sgn_sgn @ A @ X ) @ ( sgn_sgn @ A @ Y ) ) ) ) ).
% Real_Vector_Spaces.sgn_mult
thf(fact_3446_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_mult
thf(fact_3447_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_3448_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_3449_sgn__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% sgn_eq_0_iff
thf(fact_3450_sgn__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( sgn_sgn @ A @ X )
= ( zero_zero @ A ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% sgn_zero_iff
thf(fact_3451_sgn__not__eq__imp,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A3: A] :
( ( ( sgn_sgn @ A @ B2 )
!= ( sgn_sgn @ A @ A3 ) )
=> ( ( ( sgn_sgn @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( sgn_sgn @ A @ B2 )
!= ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ B2 ) ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_3452_sgn__minus__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% sgn_minus_1
thf(fact_3453_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A )
= ( ^ [K3: A] : ( times_times @ A @ K3 @ ( sgn_sgn @ A @ K3 ) ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_3454_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= A3 ) ) ).
% abs_mult_sgn
thf(fact_3455_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= A3 ) ) ).
% sgn_mult_abs
thf(fact_3456_mult__sgn__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ X ) @ ( abs_abs @ A @ X ) )
= X ) ) ).
% mult_sgn_abs
thf(fact_3457_int__sgnE,axiom,
! [K2: int] :
~ ! [N3: nat,L4: int] :
( K2
!= ( times_times @ int @ ( sgn_sgn @ int @ L4 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% int_sgnE
thf(fact_3458_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_3459_div__eq__sgn__abs,axiom,
! [K2: int,L: int] :
( ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ K2 @ L )
= ( divide_divide @ int @ ( abs_abs @ int @ K2 ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_eq_sgn_abs
thf(fact_3460_frac__ge__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) ) ) ).
% frac_ge_0
thf(fact_3461_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_3462_frac__1__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( archimedean_frac @ A @ X ) ) ) ).
% frac_1_eq
thf(fact_3463_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_1_pos
thf(fact_3464_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_3465_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X4: A] :
( if @ A
@ ( X4
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X4 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_3466_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_3467_norm__sgn,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( zero_zero @ real ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( one_one @ real ) ) ) ) ) ).
% norm_sgn
thf(fact_3468_div__sgn__abs__cancel,axiom,
! [V3: int,K2: int,L: int] :
( ( V3
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ V3 ) @ ( abs_abs @ int @ K2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ V3 ) @ ( abs_abs @ int @ L ) ) )
= ( divide_divide @ int @ ( abs_abs @ int @ K2 ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_3469_div__dvd__sgn__abs,axiom,
! [L: int,K2: int] :
( ( dvd_dvd @ int @ L @ K2 )
=> ( ( divide_divide @ int @ K2 @ L )
= ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( sgn_sgn @ int @ L ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K2 ) @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_dvd_sgn_abs
thf(fact_3470_frac__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_frac @ A )
= ( ^ [X4: A] : ( minus_minus @ A @ X4 @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X4 ) ) ) ) ) ) ).
% frac_def
thf(fact_3471_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= X )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_3472_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_3473_eucl__rel__int__remainderI,axiom,
! [R3: int,L: int,K2: int,Q2: int] :
( ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ L ) )
=> ( ( K2
= ( plus_plus @ int @ ( times_times @ int @ Q2 @ L ) @ R3 ) )
=> ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q2 @ R3 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_3474_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A13: int,A24: int,A32: product_prod @ int @ int] :
( ? [K3: int] :
( ( A13 = K3 )
& ( A24
= ( zero_zero @ int ) )
& ( A32
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L2: int,K3: int,Q5: int] :
( ( A13 = K3 )
& ( A24 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q5 @ L2 ) ) )
| ? [R: int,L2: int,K3: int,Q5: int] :
( ( A13 = K3 )
& ( A24 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q5 @ R ) )
& ( ( sgn_sgn @ int @ R )
= ( sgn_sgn @ int @ L2 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ L2 ) )
& ( K3
= ( plus_plus @ int @ ( times_times @ int @ Q5 @ L2 ) @ R ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_3475_eucl__rel__int_Ocases,axiom,
! [A12: int,A23: int,A33: product_prod @ int @ int] :
( ( eucl_rel_int @ A12 @ A23 @ A33 )
=> ( ( ( A23
= ( zero_zero @ int ) )
=> ( A33
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A12 ) ) )
=> ( ! [Q3: int] :
( ( A33
= ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) )
=> ( ( A23
!= ( zero_zero @ int ) )
=> ( A12
!= ( times_times @ int @ Q3 @ A23 ) ) ) )
=> ~ ! [R4: int,Q3: int] :
( ( A33
= ( product_Pair @ int @ int @ Q3 @ R4 ) )
=> ( ( ( sgn_sgn @ int @ R4 )
= ( sgn_sgn @ int @ A23 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R4 ) @ ( abs_abs @ int @ A23 ) )
=> ( A12
!= ( plus_plus @ int @ ( times_times @ int @ Q3 @ A23 ) @ R4 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_3476_div__noneq__sgn__abs,axiom,
! [L: int,K2: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( ( sgn_sgn @ int @ K2 )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ K2 @ L )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ ( abs_abs @ int @ K2 ) @ ( abs_abs @ int @ L ) ) )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L @ K2 ) ) ) ) ) ) ).
% div_noneq_sgn_abs
thf(fact_3477_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_3478_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_3479_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_3480_modulo__int__def,axiom,
( ( modulo_modulo @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L2 ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L2 )
@ ( minus_minus @ int
@ ( times_times @ int @ ( abs_abs @ int @ L2 )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L2 @ K3 ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_3481_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_3482_num_Osize__gen_I3_J,axiom,
! [X33: num] :
( ( size_num @ ( bit1 @ X33 ) )
= ( plus_plus @ nat @ ( size_num @ X33 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(3)
thf(fact_3483_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N5: nat,A7: A] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_3484_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% mask_nat_positive_iff
thf(fact_3485_take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% take_bit_of_0
thf(fact_3486_take__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% take_bit_0
thf(fact_3487_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_3488_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_3489_mask__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( ( bit_se2239418461657761734s_mask @ A @ N )
= ( zero_zero @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% mask_eq_0_iff
thf(fact_3490_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_zero @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_3491_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_3492_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_3493_nat__0__iff,axiom,
! [I2: int] :
( ( ( nat2 @ I2 )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_3494_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z3 )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_3495_nat__neg__numeral,axiom,
! [K2: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_3496_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_3497_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_3498_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_3499_take__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% take_bit_of_1
thf(fact_3500_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L ) @ ( one_one @ int ) ) ) ) ).
% card_atLeastAtMost_int
thf(fact_3501_nat__ceiling__le__eq,axiom,
! [X: real,A3: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A3 )
= ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A3 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_3502_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_3503_one__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z3 ) ) ).
% one_less_nat_eq
thf(fact_3504_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_3505_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_3506_nat__le__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_3507_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_3508_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_3509_take__bit__tightened__less__eq__nat,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M2 @ Q2 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N @ Q2 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_3510_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_3511_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_3512_take__bit__mult,axiom,
! [N: nat,K2: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ K2 @ L ) ) ) ).
% take_bit_mult
thf(fact_3513_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% less_eq_mask
thf(fact_3514_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_3515_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_3516_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_3517_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ X @ Y )
=> ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_3518_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_3519_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_3520_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_3521_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_3522_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_3523_of__nat__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R3: A] : ( ord_less_eq @ A @ R3 @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R3 ) ) ) ) ) ).
% of_nat_ceiling
thf(fact_3524_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_le_iff
thf(fact_3525_nat__abs__mult__distrib,axiom,
! [W2: int,Z3: int] :
( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W2 @ Z3 ) ) )
= ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W2 ) ) @ ( nat2 @ ( abs_abs @ int @ Z3 ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_3526_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A7: nat,B5: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_3527_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_3528_of__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R3 )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archim6421214686448440834_floor @ A @ R3 ) ) ) @ R3 ) ) ) ).
% of_nat_floor
thf(fact_3529_nat__le__eq__zle,axiom,
! [W2: int,Z3: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W2 )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq @ int @ W2 @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_3530_nat__eq__iff2,axiom,
! [M2: nat,W2: int] :
( ( M2
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_3531_nat__eq__iff,axiom,
! [W2: int,M2: nat] :
( ( ( nat2 @ W2 )
= M2 )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_3532_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_3533_split__nat,axiom,
! [P2: nat > $o,I2: int] :
( ( P2 @ ( nat2 @ I2 ) )
= ( ! [N5: nat] :
( ( I2
= ( semiring_1_of_nat @ int @ N5 ) )
=> ( P2 @ N5 ) )
& ( ( ord_less @ int @ I2 @ ( zero_zero @ int ) )
=> ( P2 @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_3534_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_3535_nat__mult__distrib,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( nat2 @ ( times_times @ int @ Z3 @ Z6 ) )
= ( times_times @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ).
% nat_mult_distrib
thf(fact_3536_Suc__as__int,axiom,
( suc
= ( ^ [A7: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( one_one @ int ) ) ) ) ) ).
% Suc_as_int
thf(fact_3537_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_3538_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= ( zero_zero @ nat ) ) ) ).
% nat_floor_neg
thf(fact_3539_div__abs__eq__div__nat,axiom,
! [K2: int,L: int] :
( ( divide_divide @ int @ ( abs_abs @ int @ K2 ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_3540_floor__eq3,axiom,
! [N: nat,X: real] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N ) ) ) ).
% floor_eq3
thf(fact_3541_le__nat__floor,axiom,
! [X: nat,A3: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X ) @ A3 )
=> ( ord_less_eq @ nat @ X @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A3 ) ) ) ) ).
% le_nat_floor
thf(fact_3542_mod__abs__eq__div__nat,axiom,
! [K2: int,L: int] :
( ( modulo_modulo @ int @ ( abs_abs @ int @ K2 ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% mod_abs_eq_div_nat
thf(fact_3543_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_3544_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_3545_sgn__power__injE,axiom,
! [A3: real,N: nat,X: real,B2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A3 ) @ ( power_power @ real @ ( abs_abs @ real @ A3 ) @ N ) )
= X )
=> ( ( X
= ( times_times @ real @ ( sgn_sgn @ real @ B2 ) @ ( power_power @ real @ ( abs_abs @ real @ B2 ) @ N ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A3 = B2 ) ) ) ) ).
% sgn_power_injE
thf(fact_3546_Suc__nat__eq__nat__zadd1,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( suc @ ( nat2 @ Z3 ) )
= ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_3547_nat__mult__distrib__neg,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z3 @ Z6 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z6 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_3548_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_3549_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N ) ) ) ).
% floor_eq4
thf(fact_3550_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_3551_diff__nat__eq__if,axiom,
! [Z6: int,Z3: int] :
( ( ( ord_less @ int @ Z6 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) )
= ( nat2 @ Z3 ) ) )
& ( ~ ( ord_less @ int @ Z6 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z3 @ Z6 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z3 @ Z6 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_3552_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_3553_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_3554_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_3555_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_3556_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_3557_nat__dvd__iff,axiom,
! [Z3: int,M2: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ Z3 ) @ M2 )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( dvd_dvd @ int @ Z3 @ ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_dvd_iff
thf(fact_3558_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_3559_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_3560_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_3561_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_3562_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_3563_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_3564_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N5: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N5 ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_3565_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_3566_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_3567_arctan__inverse,axiom,
! [X: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ X ) )
= ( minus_minus @ real @ ( divide_divide @ real @ ( times_times @ real @ ( sgn_sgn @ real @ X ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( arctan @ X ) ) ) ) ).
% arctan_inverse
thf(fact_3568_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_3569_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_3570_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_3571_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_3572_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_3573_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_3574_or_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% or.left_neutral
thf(fact_3575_or_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% or.right_neutral
thf(fact_3576_real__sqrt__mult__self,axiom,
! [A3: real] :
( ( times_times @ real @ ( sqrt @ A3 ) @ ( sqrt @ A3 ) )
= ( abs_abs @ real @ A3 ) ) ).
% real_sqrt_mult_self
thf(fact_3577_real__sqrt__abs2,axiom,
! [X: real] :
( ( sqrt @ ( times_times @ real @ X @ X ) )
= ( abs_abs @ real @ X ) ) ).
% real_sqrt_abs2
thf(fact_3578_pred__numeral__inc,axiom,
! [K2: num] :
( ( pred_numeral @ ( inc @ K2 ) )
= ( numeral_numeral @ nat @ K2 ) ) ).
% pred_numeral_inc
thf(fact_3579_or__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% or_numerals(3)
thf(fact_3580_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_3581_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_3582_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_3583_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_3584_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X: real,Y: real,Xa2: real,Ya: real] :
( ( power_power @ real @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ 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 @ X @ ( 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_3585_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_3586_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_3587_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_3588_real__sqrt__mult,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( times_times @ real @ X @ Y ) )
= ( times_times @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_mult
thf(fact_3589_or__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A] :
( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% or_eq_0_iff
thf(fact_3590_bit_Odisj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( zero_zero @ A ) )
= X ) ) ).
% bit.disj_zero_right
thf(fact_3591_num__induct,axiom,
! [P2: num > $o,X: num] :
( ( P2 @ one2 )
=> ( ! [X3: num] :
( ( P2 @ X3 )
=> ( P2 @ ( inc @ X3 ) ) )
=> ( P2 @ X ) ) ) ).
% num_induct
thf(fact_3592_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus @ num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus @ num @ X @ Y ) ) ) ).
% add_inc
thf(fact_3593_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_3594_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_3595_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_3596_add__One,axiom,
! [X: num] :
( ( plus_plus @ num @ X @ one2 )
= ( inc @ X ) ) ).
% add_One
thf(fact_3597_le__real__sqrt__sumsq,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( times_times @ real @ X @ X ) @ ( times_times @ real @ Y @ Y ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_3598_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_3599_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_3600_mult__inc,axiom,
! [X: num,Y: num] :
( ( times_times @ num @ X @ ( inc @ Y ) )
= ( plus_plus @ num @ ( times_times @ num @ X @ Y ) @ X ) ) ).
% mult_inc
thf(fact_3601_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_3602_numeral__inc,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( numeral_numeral @ A @ ( inc @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% numeral_inc
thf(fact_3603_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S3 ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ).
% sup_Un_eq2
thf(fact_3604_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S3 ) ) )
= ( R2 = S3 ) ) ).
% pred_equals_eq2
thf(fact_3605_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_3606_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S3 ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ).
% inf_Int_eq2
thf(fact_3607_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_3608_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_3609_subrelI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ).
% subrelI
thf(fact_3610_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X: real,Y: real,Xa2: real,Ya: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_ge_zero
thf(fact_3611_arith__geo__mean__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( times_times @ real @ X @ Y ) ) @ ( divide_divide @ real @ ( plus_plus @ real @ X @ Y ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_3612_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_3613_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N5: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N5 ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N5 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N5 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_3614_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_3615_cis__2pi,axiom,
( ( cis @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ complex ) ) ).
% cis_2pi
thf(fact_3616_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_3617_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_3618_bit__0__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( zero_zero @ A ) )
= ( bot_bot @ ( nat > $o ) ) ) ) ).
% bit_0_eq
thf(fact_3619_or__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(4)
thf(fact_3620_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_3621_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_3622_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_3623_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_3624_or__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(3)
thf(fact_3625_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_3626_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_3627_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_3628_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_3629_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
~ ( bot_bot @ ( A > B > $o ) @ X @ Y ) ).
% bot2E
thf(fact_3630_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_3631_bit__1__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% bit_1_iff
thf(fact_3632_bit__of__bool__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [B2: $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ N )
= ( B2
& ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% bit_of_bool_iff
thf(fact_3633_cis__mult,axiom,
! [A3: real,B2: real] :
( ( times_times @ complex @ ( cis @ A3 ) @ ( cis @ B2 ) )
= ( cis @ ( plus_plus @ real @ A3 @ B2 ) ) ) ).
% cis_mult
thf(fact_3634_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_3635_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_3636_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_3637_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_3638_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_3639_int__bit__bound,axiom,
! [K2: int] :
~ ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ N3 @ M3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K2 @ M3 )
= ( bit_se5641148757651400278ts_bit @ int @ K2 @ N3 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N3 ) ) ) ) ) ).
% int_bit_bound
thf(fact_3640_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_3641_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_3642_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_3643_set__bit__eq,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N5: nat,K3: int] :
( plus_plus @ int @ K3
@ ( times_times @ int
@ ( zero_neq_one_of_bool @ int
@ ~ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N5 ) )
@ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ).
% set_bit_eq
thf(fact_3644_unset__bit__eq,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N5: nat,K3: int] : ( minus_minus @ int @ K3 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N5 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_3645_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A,N: nat] :
( ! [J3: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J3 ) )
=> ( ( 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_3646_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A7: A,N5: nat] :
( ( ( N5
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A7 ) )
& ( ( N5
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_3647_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_3648_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_3649_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M6: nat,N5: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M6 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_3650_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M6: nat,N5: nat] :
( if @ nat
@ ( M6
= ( zero_zero @ nat ) )
@ N5
@ ( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ M6
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_3651_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_3652_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_3653_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_3654_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_3655_frac__in__Ints__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( member @ A @ ( archimedean_frac @ A @ X ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_in_Ints_iff
thf(fact_3656_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y8: B] :
( ( X = X9 )
& ( Y = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% The_split_eq
thf(fact_3657_frac__eq__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= ( zero_zero @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_eq_0_iff
thf(fact_3658_floor__add2,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
| ( member @ A @ Y @ ( ring_1_Ints @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ) ).
% floor_add2
thf(fact_3659_frac__gt__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) )
= ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ) ).
% frac_gt_0_iff
thf(fact_3660_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_3661_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_3662_finite__same__card__bij,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ( finite_card @ A @ A6 )
= ( finite_card @ B @ B6 ) )
=> ? [H4: A > B] : ( bij_betw @ A @ B @ H4 @ A6 @ B6 ) ) ) ) ).
% finite_same_card_bij
thf(fact_3663_bij__betw__iff__card,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ? [F4: A > B] : ( bij_betw @ A @ B @ F4 @ A6 @ B6 ) )
= ( ( finite_card @ A @ A6 )
= ( finite_card @ B @ B6 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_3664_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_3665_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_3666_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_3667_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_3668_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_3669_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_3670_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 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_3671_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 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_3672_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_3673_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M5: set @ A] :
( ( finite_finite2 @ A @ M5 )
=> ? [H4: nat > A] : ( bij_betw @ nat @ A @ H4 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M5 ) ) @ M5 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_3674_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ B,T5: set @ C,H: B > C,S3: set @ B,T6: set @ C,G3: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S5 )
=> ( ( G3 @ ( H @ A5 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( G3 @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( G3 @ ( H @ X4 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ G3 @ T6 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_3675_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_3676_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_3677_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_3678_sin__times__pi__eq__0,axiom,
! [X: real] :
( ( ( sin @ real @ ( times_times @ real @ X @ pi ) )
= ( zero_zero @ real ) )
= ( member @ real @ X @ ( ring_1_Ints @ real ) ) ) ).
% sin_times_pi_eq_0
thf(fact_3679_bij__betw__nth,axiom,
! [A: $tType,Xs2: list @ A,A6: set @ nat,B6: set @ A] :
( ( distinct @ A @ Xs2 )
=> ( ( A6
= ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
=> ( ( B6
= ( set2 @ A @ Xs2 ) )
=> ( bij_betw @ nat @ A @ ( nth @ A @ Xs2 ) @ A6 @ B6 ) ) ) ) ).
% bij_betw_nth
thf(fact_3680_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X ) ) ) ) ) ) ).
% frac_neg
thf(fact_3681_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_3682_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: A] :
( ( ( archimedean_frac @ A @ X )
= A3 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A3 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_3683_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_3684_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_3685_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_3686_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_3687_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_3688_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_3689_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_3690_bit_Oxor__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ X )
= ( zero_zero @ A ) ) ) ).
% bit.xor_self
thf(fact_3691_xor__self__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% xor_self_eq
thf(fact_3692_xor_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% xor.left_neutral
thf(fact_3693_xor_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% xor.right_neutral
thf(fact_3694_complex__i__mult__minus,axiom,
! [X: complex] :
( ( times_times @ complex @ imaginary_unit @ ( times_times @ complex @ imaginary_unit @ X ) )
= ( uminus_uminus @ complex @ X ) ) ).
% complex_i_mult_minus
thf(fact_3695_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F3: B > A,A3: A,X: B,Xs2: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ ( cons @ B @ X @ Xs2 ) )
= ( plus_plus @ A @ ( F3 @ X ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ Xs2 ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_3696_divide__numeral__i,axiom,
! [Z3: complex,N: num] :
( ( divide_divide @ complex @ Z3 @ ( times_times @ complex @ ( numeral_numeral @ complex @ N ) @ imaginary_unit ) )
= ( divide_divide @ complex @ ( uminus_uminus @ complex @ ( times_times @ complex @ imaginary_unit @ Z3 ) ) @ ( numeral_numeral @ complex @ N ) ) ) ).
% divide_numeral_i
thf(fact_3697_divide__i,axiom,
! [X: complex] :
( ( divide_divide @ complex @ X @ imaginary_unit )
= ( times_times @ complex @ ( uminus_uminus @ complex @ imaginary_unit ) @ X ) ) ).
% divide_i
thf(fact_3698_i__squared,axiom,
( ( times_times @ complex @ imaginary_unit @ imaginary_unit )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% i_squared
thf(fact_3699_xor__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(3)
thf(fact_3700_xor__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(7)
thf(fact_3701_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_3702_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_3703_xor__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% xor_nat_numerals(3)
thf(fact_3704_xor__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ X ) ) ) ).
% xor_nat_numerals(4)
thf(fact_3705_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_3706_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_3707_i__times__eq__iff,axiom,
! [W2: complex,Z3: complex] :
( ( ( times_times @ complex @ imaginary_unit @ W2 )
= Z3 )
= ( W2
= ( uminus_uminus @ complex @ ( times_times @ complex @ imaginary_unit @ Z3 ) ) ) ) ).
% i_times_eq_iff
thf(fact_3708_i__mult__Complex,axiom,
! [A3: real,B2: real] :
( ( times_times @ complex @ imaginary_unit @ ( complex2 @ A3 @ B2 ) )
= ( complex2 @ ( uminus_uminus @ real @ B2 ) @ A3 ) ) ).
% i_mult_Complex
thf(fact_3709_Complex__mult__i,axiom,
! [A3: real,B2: real] :
( ( times_times @ complex @ ( complex2 @ A3 @ B2 ) @ imaginary_unit )
= ( complex2 @ ( uminus_uminus @ real @ B2 ) @ A3 ) ) ).
% Complex_mult_i
thf(fact_3710_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M6: nat,N5: nat] :
( if @ nat
@ ( M6
= ( zero_zero @ nat ) )
@ N5
@ ( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ M6
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_3711_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M6: nat,N5: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M6 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_3712_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_3713_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_3714_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F5: set @ A,I5: set @ A,F3: A > B,I2: A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I: A] :
( ( member @ A @ I @ I5 )
& ( ( F3 @ I )
!= ( zero_zero @ B ) ) ) )
@ F5 )
=> ( ( ( member @ A @ I2 @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I5 ) @ ( F3 @ I2 ) ) ) )
& ( ~ ( member @ A @ I2 @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I5 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_3715_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
@ ^ [S6: A] :
( ( member @ A @ S6 @ S3 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ S6 ) ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_3716_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_3717_Least__eq__0,axiom,
! [P2: nat > $o] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P2 )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_3718_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_3719_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I2 )
& ( ord_less @ A @ I2 @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_3720_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_3721_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I2: A,J2: A,M2: A,N: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ J2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ N ) )
= ( ( ord_less_eq @ A @ J2 @ I2 )
| ( ( ord_less_eq @ A @ M2 @ I2 )
& ( ord_less_eq @ A @ J2 @ N ) ) ) ) ) ).
% ivl_subset
thf(fact_3722_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_3723_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_3724_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ico_iff
thf(fact_3725_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I2: A,N: A,M2: A] :
( ( ord_less_eq @ A @ I2 @ N )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ N ) )
= ( set_or7035219750837199246ssThan @ A @ N @ M2 ) ) ) ) ).
% ivl_diff
thf(fact_3726_lessThan__minus__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: A,M2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_ord_lessThan @ A @ N ) @ ( set_ord_lessThan @ A @ M2 ) )
= ( set_or7035219750837199246ssThan @ A @ M2 @ N ) ) ) ).
% lessThan_minus_lessThan
thf(fact_3727_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_3728_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_3729_atLeastLessThan__singleton,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ M2 ) )
= ( insert @ nat @ M2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_3730_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,P: B > A,I2: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( P @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I2 @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P @ ( insert @ B @ I2 @ I5 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P @ I5 ) ) )
& ( ~ ( member @ B @ I2 @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P @ ( insert @ B @ I2 @ I5 ) )
= ( plus_plus @ A @ ( P @ I2 ) @ ( groups1027152243600224163dd_sum @ B @ A @ P @ I5 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_3731_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_3732_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_3733_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,A3: A,Q: A > $o] :
( ( P2 @ A3 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P2 ) ) ) ) ) ).
% LeastI2
thf(fact_3734_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o] :
( ? [X_1: A] : ( P2 @ X_1 )
=> ( P2 @ ( ord_Least @ A @ P2 ) ) ) ) ).
% LeastI_ex
thf(fact_3735_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P2 @ X_1 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P2 ) ) ) ) ) ).
% LeastI2_ex
thf(fact_3736_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,K2: A] :
( ( P2 @ K2 )
=> ( P2 @ ( ord_Least @ A @ P2 ) ) ) ) ).
% LeastI
thf(fact_3737_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
= ( ( A3 = C3 )
& ( B2 = D3 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_3738_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( A3 = C3 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_3739_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( B2 = D3 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_3740_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P2 @ X_1 )
=> ( ! [A5: A] :
( ( P2 @ A5 )
=> ( ! [B7: A] :
( ( P2 @ B7 )
=> ( ord_less_eq @ A @ A5 @ B7 ) )
=> ( Q @ A5 ) ) )
=> ( Q @ ( ord_Least @ A @ P2 ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_3741_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,A3: A,Q: A > $o] :
( ( P2 @ A3 )
=> ( ! [A5: A] :
( ( P2 @ A5 )
=> ( ! [B7: A] :
( ( P2 @ B7 )
=> ( ord_less_eq @ A @ A5 @ B7 ) )
=> ( Q @ A5 ) ) )
=> ( Q @ ( ord_Least @ A @ P2 ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_3742_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X: A] :
( ( P2 @ X )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( ord_Least @ A @ P2 )
= X ) ) ) ) ).
% Least_equality
thf(fact_3743_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X: A,Q: A > $o] :
( ( P2 @ X )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( ! [Y6: A] :
( ( P2 @ Y6 )
=> ( ord_less_eq @ A @ X3 @ Y6 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Least @ A @ P2 ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_3744_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,Z3: A] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ X5 @ Y3 ) )
& ! [Y3: A] :
( ( ( P2 @ Y3 )
& ! [Ya2: A] :
( ( P2 @ Ya2 )
=> ( ord_less_eq @ A @ Y3 @ Ya2 ) ) )
=> ( Y3 = X5 ) ) )
=> ( ( P2 @ Z3 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P2 ) @ Z3 ) ) ) ) ).
% Least1_le
thf(fact_3745_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ X5 @ Y3 ) )
& ! [Y3: A] :
( ( ( P2 @ Y3 )
& ! [Ya2: A] :
( ( P2 @ Ya2 )
=> ( ord_less_eq @ A @ Y3 @ Ya2 ) ) )
=> ( Y3 = X5 ) ) )
=> ( P2 @ ( ord_Least @ A @ P2 ) ) ) ) ).
% Least1I
thf(fact_3746_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,I5: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( G3 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ I5 ) ) ) ).
% sum.non_neutral'
thf(fact_3747_Least__le,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,K2: A] :
( ( P2 @ K2 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P2 ) @ K2 ) ) ) ).
% Least_le
thf(fact_3748_not__less__Least,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [K2: A,P2: A > $o] :
( ( ord_less @ A @ K2 @ ( ord_Least @ A @ P2 ) )
=> ~ ( P2 @ K2 ) ) ) ).
% not_less_Least
thf(fact_3749_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_3750_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ico
thf(fact_3751_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_3752_all__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M6: nat] :
( ( ord_less @ nat @ M6 @ N )
=> ( P2 @ M6 ) ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P2 @ X4 ) ) ) ) ).
% all_nat_less_eq
thf(fact_3753_ex__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M6: nat] :
( ( ord_less @ nat @ M6 @ N )
& ( P2 @ M6 ) ) )
= ( ? [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P2 @ X4 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_3754_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_3755_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_3756_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_3757_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_3758_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
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_3759_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
@ ^ [I: nat] : ( G3 @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_3760_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
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_3761_Least__Suc2,axiom,
! [P2: nat > $o,N: nat,Q: nat > $o,M2: nat] :
( ( P2 @ N )
=> ( ( Q @ M2 )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ( ! [K: nat] :
( ( P2 @ ( suc @ K ) )
= ( Q @ K ) )
=> ( ( ord_Least @ nat @ P2 )
= ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_3762_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
@ ^ [I: nat] : ( G3 @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_3763_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 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C3 @ X3 )
=> ( ( ord_less @ B @ X3 @ D3 )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) ) )
=> ( ( 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_3764_Least__Suc,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P2 )
= ( suc
@ ( ord_Least @ nat
@ ^ [M6: nat] : ( P2 @ ( suc @ M6 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_3765_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 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C3 @ X3 )
=> ( ( ord_less @ B @ X3 @ D3 )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) ) )
=> ( ( 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_3766_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_3767_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_3768_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M5: set @ A] :
( ( finite_finite2 @ A @ M5 )
=> ? [H4: A > nat] : ( bij_betw @ A @ nat @ H4 @ M5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M5 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_3769_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,P: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_3770_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,P: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_3771_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_3772_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_3773_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,P: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_3774_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_3775_enumerate__0,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A] :
( ( infini527867602293511546merate @ A @ S3 @ ( zero_zero @ nat ) )
= ( ord_Least @ A
@ ^ [N5: A] : ( member @ A @ N5 @ S3 ) ) ) ) ).
% enumerate_0
thf(fact_3776_subset__eq__atLeast0__lessThan__finite,axiom,
! [N7: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_3777_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T6: set @ B,G3: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T6 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_3778_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T6: set @ B,H: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ T6 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_3779_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T6: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T6 )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_3780_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T6: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T6 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_3781_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_3782_atLeastLessThan__add__Un,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( set_or7035219750837199246ssThan @ nat @ I2 @ ( plus_plus @ nat @ J2 @ K2 ) )
= ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I2 @ J2 ) @ ( set_or7035219750837199246ssThan @ nat @ J2 @ ( plus_plus @ nat @ J2 @ K2 ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_3783_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_3784_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_3785_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_3786_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_3787_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_3788_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_3789_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_3790_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_3791_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
@ ^ [I: B] : ( plus_plus @ A @ ( G3 @ I ) @ ( H @ I ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G3 @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H @ I5 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_3792_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A7: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A7 @ B5 ) @ ( insert @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_3793_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_3794_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_3795_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P5: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I7 )
& ( ( P5 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P5
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I7 )
& ( ( P5 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_3796_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_3797_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_3798_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M5: set @ A] :
( ( finite_finite2 @ A @ M5 )
=> ? [H4: nat > A] : ( bij_betw @ nat @ A @ H4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M5 ) ) @ M5 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_3799_atLeastLessThanSuc,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
= ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_3800_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
@ ^ [I: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I ) ) @ ( F3 @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M2 ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_3801_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
@ ^ [I: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_3802_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( A3 @ I @ J )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_3803_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
@ ^ [I: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_3804_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( A3 @ I @ J )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_3805_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M6: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M6 @ K2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ M6 @ 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_3806_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M6: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M6 @ K2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ M6 @ 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_3807_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_3808_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_3809_subset__eq__atLeast0__lessThan__card,axiom,
! [N7: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N7 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_3810_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F3: ( A > B ) > C,G3: C] :
( ( F3
= ( ^ [X4: A > B] : G3 ) )
=> ( ( F3
@ ^ [X4: A] : ( zero_zero @ B ) )
= G3 ) ) ) ).
% fun_cong_unused_0
thf(fact_3811_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_3812_ivl__disj__un__singleton_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_3813_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_3814_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_3815_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N5: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_3816_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
@ ^ [I: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3817_atLeastLessThan__nat__numeral,axiom,
! [M2: nat,K2: num] :
( ( ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K2 ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K2 ) )
= ( insert @ nat @ ( pred_numeral @ K2 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( pred_numeral @ K2 ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K2 ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K2 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_3818_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
@ ^ [I: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3819_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A7: A,N5: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_3820_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N5: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N5 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_3821_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
@ ^ [S6: A] :
( ( member @ A @ S6 @ S3 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ S6 ) ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_3822_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] :
! [M6: nat] :
( ( ord_less_eq @ nat @ N6 @ M6 )
=> ! [N5: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_or7035219750837199246ssThan @ nat @ M6 @ N5 ) ) ) @ E4 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_3823_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A,S2: A,K2: nat] :
( ( sums @ A @ F3 @ S2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( sums @ A
@ ^ [N5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N5 @ K2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ N5 @ K2 ) @ K2 ) ) )
@ S2 ) ) ) ) ).
% sums_group
thf(fact_3824_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_3825_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_3826_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S3
@ ( insert @ A
@ ( ord_Least @ A
@ ^ [N5: A] : ( member @ A @ N5 @ S3 ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N ) ) ) ).
% enumerate_Suc
thf(fact_3827_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
@ ^ [I: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K2 @ I ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_3828_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A7: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A7 @ ( semiring_1_of_nat @ A @ I ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_3829_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K2: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K2 ) @ ( semiring_char_0_fact @ A @ K2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_3830_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A3: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( gbinomial @ A @ A3 @ K2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_3831_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A7: A,K3: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( minus_minus @ A @ A7 @ ( semiring_1_of_nat @ A @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_3832_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I5: set @ A,F3: A > B,I2: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I: A] :
( ( member @ A @ I @ I5 )
& ( ( F3 @ I )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I2 @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I5 ) @ ( F3 @ I2 ) ) ) )
& ( ~ ( member @ A @ I2 @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I5 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_3833_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A7: A,Xs: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F4 @ ( nth @ B @ Xs @ N5 ) ) @ ( power_power @ A @ A7 @ N5 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_3834_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_3835_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_3836_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: nat > A,B2: nat > A] :
( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ N )
=> ( ord_less_eq @ A @ ( A3 @ I3 ) @ ( A3 @ J3 ) ) ) )
=> ( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ N )
=> ( ord_less_eq @ A @ ( B2 @ J3 ) @ ( 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_3837_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A3: nat > nat,B2: nat > nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ N )
=> ( ord_less_eq @ nat @ ( A3 @ I3 ) @ ( A3 @ J3 ) ) ) )
=> ( ! [I3: nat,J3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ N )
=> ( ord_less_eq @ nat @ ( B2 @ J3 ) @ ( B2 @ I3 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ ( A3 @ I ) @ ( B2 @ I ) )
@ ( 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_3838_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_3839_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_3840_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_3841_of__real__0,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( ( real_Vector_of_real @ A @ ( zero_zero @ real ) )
= ( zero_zero @ A ) ) ) ).
% of_real_0
thf(fact_3842_of__real__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real] :
( ( ( real_Vector_of_real @ A @ X )
= ( zero_zero @ A ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% of_real_eq_0_iff
thf(fact_3843_of__real__mult,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,Y: real] :
( ( real_Vector_of_real @ A @ ( times_times @ real @ X @ Y ) )
= ( times_times @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ).
% of_real_mult
thf(fact_3844_bit_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_right
thf(fact_3845_bit_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_left
thf(fact_3846_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_atLeastLessThan_int
thf(fact_3847_bit_Ocompl__one,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% bit.compl_one
thf(fact_3848_bit_Ocompl__zero,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( zero_zero @ A ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.compl_zero
thf(fact_3849_sin__of__real__pi,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A @ ( real_Vector_of_real @ A @ pi ) )
= ( zero_zero @ A ) ) ) ).
% sin_of_real_pi
thf(fact_3850_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_3851_exp__pi__i_H,axiom,
( ( exp @ complex @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ pi ) ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% exp_pi_i'
thf(fact_3852_exp__pi__i,axiom,
( ( exp @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ pi ) @ imaginary_unit ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% exp_pi_i
thf(fact_3853_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_3854_complex__exp__exists,axiom,
! [Z3: complex] :
? [A5: complex,R4: real] :
( Z3
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ R4 ) @ ( exp @ complex @ A5 ) ) ) ).
% complex_exp_exists
thf(fact_3855_scaleR__conv__of__real,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( ( real_V8093663219630862766scaleR @ A )
= ( ^ [R: real] : ( times_times @ A @ ( real_Vector_of_real @ A @ R ) ) ) ) ) ).
% scaleR_conv_of_real
thf(fact_3856_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_3857_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
= ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_3858_Complex__mult__complex__of__real,axiom,
! [X: real,Y: real,R3: real] :
( ( times_times @ complex @ ( complex2 @ X @ Y ) @ ( real_Vector_of_real @ complex @ R3 ) )
= ( complex2 @ ( times_times @ real @ X @ R3 ) @ ( times_times @ real @ Y @ R3 ) ) ) ).
% Complex_mult_complex_of_real
thf(fact_3859_complex__of__real__mult__Complex,axiom,
! [R3: real,X: real,Y: real] :
( ( times_times @ complex @ ( real_Vector_of_real @ complex @ R3 ) @ ( complex2 @ X @ Y ) )
= ( complex2 @ ( times_times @ real @ R3 @ X ) @ ( times_times @ real @ R3 @ Y ) ) ) ).
% complex_of_real_mult_Complex
thf(fact_3860_Complex__eq,axiom,
( complex2
= ( ^ [A7: real,B5: real] : ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ A7 ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ B5 ) ) ) ) ) ).
% Complex_eq
thf(fact_3861_cis__conv__exp,axiom,
( cis
= ( ^ [B5: real] : ( exp @ complex @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ B5 ) ) ) ) ) ).
% cis_conv_exp
thf(fact_3862_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_3863_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_3864_complex__of__real__i,axiom,
! [R3: real] :
( ( times_times @ complex @ ( real_Vector_of_real @ complex @ R3 ) @ imaginary_unit )
= ( complex2 @ ( zero_zero @ real ) @ R3 ) ) ).
% complex_of_real_i
thf(fact_3865_i__complex__of__real,axiom,
! [R3: real] :
( ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ R3 ) )
= ( complex2 @ ( zero_zero @ real ) @ R3 ) ) ).
% i_complex_of_real
thf(fact_3866_complex__split__polar,axiom,
! [Z3: complex] :
? [R4: real,A5: real] :
( Z3
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ R4 ) @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A5 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A5 ) ) ) ) ) ) ).
% complex_split_polar
thf(fact_3867_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_3868_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X )
= Y ) ) ) ) ).
% bit.compl_unique
thf(fact_3869_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_3870_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_3871_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_3872_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_3873_cos__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M2: int,X: real] :
( ( cos @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M2 ) @ ( real_Vector_of_real @ A @ X ) ) )
= ( real_Vector_of_real @ A @ ( cos @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M2 ) @ X ) ) ) ) ) ).
% cos_int_times_real
thf(fact_3874_sin__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M2: int,X: real] :
( ( sin @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M2 ) @ ( real_Vector_of_real @ A @ X ) ) )
= ( real_Vector_of_real @ A @ ( sin @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M2 ) @ X ) ) ) ) ) ).
% sin_int_times_real
thf(fact_3875_cmod__unit__one,axiom,
! [A3: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A3 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A3 ) ) ) ) )
= ( one_one @ real ) ) ).
% cmod_unit_one
thf(fact_3876_cmod__complex__polar,axiom,
! [R3: real,A3: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ R3 ) @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A3 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A3 ) ) ) ) ) )
= ( abs_abs @ real @ R3 ) ) ).
% cmod_complex_polar
thf(fact_3877_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_3878_signed__take__bit__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N5: nat,A7: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N5 @ A7 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A7 @ N5 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N5 ) ) ) ) ) ) ) ).
% signed_take_bit_def
thf(fact_3879_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_3880_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_3881_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_3882_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_3883_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( boolea3799213064322606851m_diff @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bit_se5824344971392196577ns_xor @ A ) ) ) ).
% bit.abstract_boolean_algebra_sym_diff_axioms
thf(fact_3884_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X7: nat > real] :
! [J: nat] :
? [M9: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ M9 @ M6 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ M9 @ N5 )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X7 @ M6 ) @ ( X7 @ N5 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_3885_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_3886_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_3887_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs2: list @ A,Y: nat,F3: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ Y @ ( F3 @ X ) )
=> ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F3 @ Xs2 ) ) ) ) ).
% size_list_estimation'
thf(fact_3888_size__list__pointwise,axiom,
! [A: $tType,Xs2: list @ A,F3: A > nat,G3: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ nat @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F3 @ Xs2 ) @ ( size_list @ A @ G3 @ Xs2 ) ) ) ).
% size_list_pointwise
thf(fact_3889_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,E3: real] :
( ( topolo3814608138187158403Cauchy @ A @ X8 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M8: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M8 @ M3 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ M8 @ N4 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M3 ) @ ( X8 @ N4 ) ) ) @ E3 ) ) ) ) ) ) ).
% CauchyD
thf(fact_3890_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M10: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M10 @ M )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M10 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M ) @ ( X8 @ N3 ) ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% CauchyI
thf(fact_3891_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X7: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ M9 @ M6 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ M9 @ N5 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ M6 ) @ ( X7 @ N5 ) ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_3892_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X21: A,X22: list @ A] :
( ( size_list @ A @ X @ ( cons @ A @ X21 @ X22 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X22 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_3893_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_3894_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q5: code_integer,R: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ R ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_3895_complex__div__cnj,axiom,
( ( divide_divide @ complex )
= ( ^ [A7: complex,B5: complex] : ( divide_divide @ complex @ ( times_times @ complex @ A7 @ ( cnj @ B5 ) ) @ ( real_Vector_of_real @ complex @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ B5 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_div_cnj
thf(fact_3896_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,Y4: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y4 ) )
@ ( nat_prod_decode @ N ) ) ) ) ).
% list_decode.psimps(2)
thf(fact_3897_complex__cnj__mult,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( times_times @ complex @ X @ Y ) )
= ( times_times @ complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% complex_cnj_mult
thf(fact_3898_list__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( ( nat_list_decode @ X )
= ( nat_list_decode @ Y ) )
= ( X = Y ) ) ).
% list_decode_eq
thf(fact_3899_times__integer__code_I1_J,axiom,
! [K2: code_integer] :
( ( times_times @ code_integer @ K2 @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(1)
thf(fact_3900_times__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( times_times @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(2)
thf(fact_3901_length__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs2: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs2 @ Ys ) )
=> ( ( size_size @ ( list @ A ) @ Zs )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_3902_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_3903_list__decode_Osimps_I2_J,axiom,
! [N: nat] :
( ( nat_list_decode @ ( suc @ N ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y4: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y4 ) )
@ ( nat_prod_decode @ N ) ) ) ).
% list_decode.simps(2)
thf(fact_3904_complex__mod__mult__cnj,axiom,
! [Z3: complex] :
( ( real_V7770717601297561774m_norm @ complex @ ( times_times @ complex @ Z3 @ ( cnj @ Z3 ) ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% complex_mod_mult_cnj
thf(fact_3905_complex__norm__square,axiom,
! [Z3: complex] :
( ( real_Vector_of_real @ complex @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( times_times @ complex @ Z3 @ ( cnj @ Z3 ) ) ) ).
% complex_norm_square
thf(fact_3906_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_3907_list__decode_Opelims,axiom,
! [X: nat,Y: list @ nat] :
( ( ( nat_list_decode @ X )
= Y )
=> ( ( accp @ nat @ nat_list_decode_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( nil @ nat ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( ( Y
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y4: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y4 ) )
@ ( nat_prod_decode @ N3 ) ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_3908_list__decode_Oelims,axiom,
! [X: nat,Y: list @ nat] :
( ( ( nat_list_decode @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( nil @ nat ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( Y
!= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y4: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y4 ) )
@ ( nat_prod_decode @ N3 ) ) ) ) ) ) ).
% list_decode.elims
thf(fact_3909_cnj__add__mult__eq__Re,axiom,
! [Z3: complex,W2: complex] :
( ( plus_plus @ complex @ ( times_times @ complex @ Z3 @ ( cnj @ W2 ) ) @ ( times_times @ complex @ ( cnj @ Z3 ) @ W2 ) )
= ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ ( times_times @ complex @ Z3 @ ( cnj @ W2 ) ) ) ) ) ) ).
% cnj_add_mult_eq_Re
thf(fact_3910_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_3911_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_3912_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( replicate @ A @ N @ X )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_3913_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F3: B > A,A3: A] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ ( nil @ B ) )
= ( zero_zero @ A ) ) ) ).
% horner_sum_simps(1)
thf(fact_3914_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_3915_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_3916_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs2: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs2 )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_3917_list__encode_Ocases,axiom,
! [X: list @ nat] :
( ( X
!= ( nil @ nat ) )
=> ~ ! [X3: nat,Xs3: list @ nat] :
( X
!= ( cons @ nat @ X3 @ Xs3 ) ) ) ).
% list_encode.cases
thf(fact_3918_times__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( times_times @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( times_times @ int @ Xa2 @ X ) ) ) ).
% times_integer.abs_eq
thf(fact_3919_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_3920_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs2: list @ A,Ys: list @ B,Zs: list @ C,Ws: list @ D,P2: ( 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 ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D ) @ Ws ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys5: list @ B,Z: C,Zs2: list @ C,W: D,Ws2: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys5 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys5 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys5 ) @ ( cons @ C @ Z @ Zs2 ) @ ( cons @ D @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_3921_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,Ys: list @ B,Zs: list @ C,P2: ( 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 ) @ Zs ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys5: list @ B,Z: C,Zs2: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys5 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys5 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs2 )
=> ( P2 @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys5 ) @ ( cons @ C @ Z @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_3922_list__induct2,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,P2: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys5: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys5 ) )
=> ( ( P2 @ Xs3 @ Ys5 )
=> ( P2 @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys5 ) ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_3923_replicate__0,axiom,
! [A: $tType,X: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_3924_list_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_list @ A @ X @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size_gen(1)
thf(fact_3925_scaleR__complex_Osimps_I1_J,axiom,
! [R3: real,X: complex] :
( ( re @ ( real_V8093663219630862766scaleR @ complex @ R3 @ X ) )
= ( times_times @ real @ R3 @ ( re @ X ) ) ) ).
% scaleR_complex.simps(1)
thf(fact_3926_list__decode_Osimps_I1_J,axiom,
( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% list_decode.simps(1)
thf(fact_3927_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y: A] :
( ( count_list @ A @ ( nil @ A ) @ Y )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_3928_Re__complex__div__eq__0,axiom,
! [A3: complex,B2: complex] :
( ( ( re @ ( divide_divide @ complex @ A3 @ B2 ) )
= ( zero_zero @ real ) )
= ( ( re @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) )
= ( zero_zero @ real ) ) ) ).
% Re_complex_div_eq_0
thf(fact_3929_complex__mod__sqrt__Re__mult__cnj,axiom,
( ( real_V7770717601297561774m_norm @ complex )
= ( ^ [Z4: complex] : ( sqrt @ ( re @ ( times_times @ complex @ Z4 @ ( cnj @ Z4 ) ) ) ) ) ) ).
% complex_mod_sqrt_Re_mult_cnj
thf(fact_3930_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A12: list @ A,A23: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A12 @ A23 ) @ ( listrel @ A @ B @ R3 ) )
=> ( ( ( A12
= ( nil @ A ) )
=> ( A23
!= ( nil @ B ) ) )
=> ~ ! [X3: A,Y3: B,Xs3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
=> ! [Ys5: list @ B] :
( ( A23
= ( cons @ B @ Y3 @ Ys5 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys5 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_3931_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A12: list @ A,A23: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A12 @ A23 ) @ ( listrel @ A @ B @ R3 ) )
= ( ( ( A12
= ( nil @ A ) )
& ( A23
= ( nil @ B ) ) )
| ? [X4: A,Y4: B,Xs: list @ A,Ys3: list @ B] :
( ( A12
= ( cons @ A @ X4 @ Xs ) )
& ( A23
= ( cons @ B @ Y4 @ Ys3 ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R3 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys3 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel.simps
thf(fact_3932_Re__complex__div__gt__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A3 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) ) ) ).
% Re_complex_div_gt_0
thf(fact_3933_Re__complex__div__lt__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less @ real @ ( re @ ( divide_divide @ complex @ A3 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_lt_0
thf(fact_3934_Re__complex__div__le__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less_eq @ real @ ( re @ ( divide_divide @ complex @ A3 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_le_0
thf(fact_3935_Re__complex__div__ge__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A3 @ B2 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) ) ) ).
% Re_complex_div_ge_0
thf(fact_3936_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_3937_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_3938_complex__add__cnj,axiom,
! [Z3: complex] :
( ( plus_plus @ complex @ Z3 @ ( cnj @ Z3 ) )
= ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ Z3 ) ) ) ) ).
% complex_add_cnj
thf(fact_3939_list__encode_Oelims,axiom,
! [X: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( ( X
= ( nil @ nat ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [X3: nat,Xs3: list @ nat] :
( ( X
= ( cons @ nat @ X3 @ Xs3 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_3940_Complex__divide,axiom,
( ( divide_divide @ complex )
= ( ^ [X4: complex,Y4: complex] : ( complex2 @ ( divide_divide @ real @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X4 ) @ ( re @ Y4 ) ) @ ( times_times @ real @ ( im @ X4 ) @ ( im @ Y4 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ ( times_times @ real @ ( im @ X4 ) @ ( re @ Y4 ) ) @ ( times_times @ real @ ( re @ X4 ) @ ( im @ Y4 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% Complex_divide
thf(fact_3941_concat__inth,axiom,
! [A: $tType,Xs2: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= X ) ).
% concat_inth
thf(fact_3942_Re__Reals__divide,axiom,
! [R3: complex,Z3: complex] :
( ( member @ complex @ R3 @ ( real_Vector_Reals @ complex ) )
=> ( ( re @ ( divide_divide @ complex @ R3 @ Z3 ) )
= ( divide_divide @ real @ ( times_times @ real @ ( re @ R3 ) @ ( re @ Z3 ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Re_Reals_divide
thf(fact_3943_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_3944_in__measures_I1_J,axiom,
! [A: $tType,X: A,Y: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% in_measures(1)
thf(fact_3945_list__encode__inverse,axiom,
! [X: list @ nat] :
( ( nat_list_decode @ ( nat_list_encode @ X ) )
= X ) ).
% list_encode_inverse
thf(fact_3946_list__decode__inverse,axiom,
! [N: nat] :
( ( nat_list_encode @ ( nat_list_decode @ N ) )
= N ) ).
% list_decode_inverse
thf(fact_3947_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_3948_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_3949_nth__append__length,axiom,
! [A: $tType,Xs2: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs2 @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_3950_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_3951_list__update__length,axiom,
! [A: $tType,Xs2: list @ A,X: A,Ys: list @ A,Y: A] :
( ( list_update @ A @ ( append @ A @ Xs2 @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) @ Y )
= ( append @ A @ Xs2 @ ( cons @ A @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_3952_complex__In__mult__cnj__zero,axiom,
! [Z3: complex] :
( ( im @ ( times_times @ complex @ Z3 @ ( cnj @ Z3 ) ) )
= ( zero_zero @ real ) ) ).
% complex_In_mult_cnj_zero
thf(fact_3953_Im__i__times,axiom,
! [Z3: complex] :
( ( im @ ( times_times @ complex @ imaginary_unit @ Z3 ) )
= ( re @ Z3 ) ) ).
% Im_i_times
thf(fact_3954_real__eq__imaginary__iff,axiom,
! [Y: complex,X: complex] :
( ( member @ complex @ Y @ ( real_Vector_Reals @ complex ) )
=> ( ( member @ complex @ X @ ( real_Vector_Reals @ complex ) )
=> ( ( X
= ( times_times @ complex @ imaginary_unit @ Y ) )
= ( ( X
= ( zero_zero @ complex ) )
& ( Y
= ( zero_zero @ complex ) ) ) ) ) ) ).
% real_eq_imaginary_iff
thf(fact_3955_imaginary__eq__real__iff,axiom,
! [Y: complex,X: complex] :
( ( member @ complex @ Y @ ( real_Vector_Reals @ complex ) )
=> ( ( member @ complex @ X @ ( real_Vector_Reals @ complex ) )
=> ( ( ( times_times @ complex @ imaginary_unit @ Y )
= X )
= ( ( X
= ( zero_zero @ complex ) )
& ( Y
= ( zero_zero @ complex ) ) ) ) ) ) ).
% imaginary_eq_real_iff
thf(fact_3956_Re__i__times,axiom,
! [Z3: complex] :
( ( re @ ( times_times @ complex @ imaginary_unit @ Z3 ) )
= ( uminus_uminus @ real @ ( im @ Z3 ) ) ) ).
% Re_i_times
thf(fact_3957_Reals__0,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_0
thf(fact_3958_list__encode__eq,axiom,
! [X: list @ nat,Y: list @ nat] :
( ( ( nat_list_encode @ X )
= ( nat_list_encode @ Y ) )
= ( X = Y ) ) ).
% list_encode_eq
thf(fact_3959_Reals__mult,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 @ ( times_times @ A @ A3 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% Reals_mult
thf(fact_3960_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_3961_nonzero__Reals__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( member @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ) ).
% nonzero_Reals_divide
thf(fact_3962_nonzero__Reals__inverse,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( member @ A @ ( inverse_inverse @ A @ A3 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% nonzero_Reals_inverse
thf(fact_3963_scaleR__complex_Osimps_I2_J,axiom,
! [R3: real,X: complex] :
( ( im @ ( real_V8093663219630862766scaleR @ complex @ R3 @ X ) )
= ( times_times @ real @ R3 @ ( im @ X ) ) ) ).
% scaleR_complex.simps(2)
thf(fact_3964_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,X3: A,Xs4: list @ A,Y3: A,Ys4: list @ A] :
( ( X3 != Y3 )
& ( Xs2
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Xs4 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_3965_list__update__append1,axiom,
! [A: $tType,I2: nat,Xs2: list @ A,Ys: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ I2 @ X )
= ( append @ A @ ( list_update @ A @ Xs2 @ I2 @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_3966_lex__append__left__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lex @ A @ R3 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R3 ) ) ) ) ).
% lex_append_left_iff
thf(fact_3967_lex__append__leftD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lex @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R3 ) ) ) ) ).
% lex_append_leftD
thf(fact_3968_lex__append__rightI,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R3: 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 @ R3 ) )
=> ( ( ( 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 @ R3 ) ) ) ) ).
% lex_append_rightI
thf(fact_3969_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs2: list @ A,R2: 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 @ R2 ) )
=> ( ( ( 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 @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_3970_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ ( nil @ nat ) )
= ( zero_zero @ nat ) ) ).
% list_encode.simps(1)
thf(fact_3971_length__Suc__conv__rev,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N ) )
= ( ? [Y4: A,Ys3: list @ A] :
( ( Xs2
= ( append @ A @ Ys3 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_3972_length__append__singleton,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_3973_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_3974_times__complex_Osimps_I2_J,axiom,
! [X: complex,Y: complex] :
( ( im @ ( times_times @ complex @ X @ Y ) )
= ( plus_plus @ real @ ( times_times @ real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times @ real @ ( im @ X ) @ ( re @ Y ) ) ) ) ).
% times_complex.simps(2)
thf(fact_3975_list__update__append,axiom,
! [A: $tType,N: nat,Xs2: list @ A,Ys: list @ A,X: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ N @ X )
= ( append @ A @ ( list_update @ A @ Xs2 @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ ( append @ A @ Xs2 @ Ys ) @ N @ X )
= ( append @ A @ Xs2 @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_3976_times__complex_Osimps_I1_J,axiom,
! [X: complex,Y: complex] :
( ( re @ ( times_times @ complex @ X @ Y ) )
= ( minus_minus @ real @ ( times_times @ real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times @ real @ ( im @ X ) @ ( im @ Y ) ) ) ) ).
% times_complex.simps(1)
thf(fact_3977_Im__complex__div__eq__0,axiom,
! [A3: complex,B2: complex] :
( ( ( im @ ( divide_divide @ complex @ A3 @ B2 ) )
= ( zero_zero @ real ) )
= ( ( im @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) )
= ( zero_zero @ real ) ) ) ).
% Im_complex_div_eq_0
thf(fact_3978_scaleR__complex_Ocode,axiom,
( ( real_V8093663219630862766scaleR @ complex )
= ( ^ [R: real,X4: complex] : ( complex2 @ ( times_times @ real @ R @ ( re @ X4 ) ) @ ( times_times @ real @ R @ ( im @ X4 ) ) ) ) ) ).
% scaleR_complex.code
thf(fact_3979_Im__Reals__divide,axiom,
! [R3: complex,Z3: complex] :
( ( member @ complex @ R3 @ ( real_Vector_Reals @ complex ) )
=> ( ( im @ ( divide_divide @ complex @ R3 @ Z3 ) )
= ( divide_divide @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( re @ R3 ) ) @ ( im @ Z3 ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Im_Reals_divide
thf(fact_3980_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_3981_Im__complex__div__lt__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less @ real @ ( im @ ( divide_divide @ complex @ A3 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_lt_0
thf(fact_3982_Im__complex__div__gt__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A3 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) ) ) ).
% Im_complex_div_gt_0
thf(fact_3983_Im__complex__div__ge__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A3 @ B2 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) ) ) ).
% Im_complex_div_ge_0
thf(fact_3984_Im__complex__div__le__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less_eq @ real @ ( im @ ( divide_divide @ complex @ A3 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_le_0
thf(fact_3985_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_3986_Re__exp,axiom,
! [Z3: complex] :
( ( re @ ( exp @ complex @ Z3 ) )
= ( times_times @ real @ ( exp @ real @ ( re @ Z3 ) ) @ ( cos @ real @ ( im @ Z3 ) ) ) ) ).
% Re_exp
thf(fact_3987_Im__exp,axiom,
! [Z3: complex] :
( ( im @ ( exp @ complex @ Z3 ) )
= ( times_times @ real @ ( exp @ real @ ( re @ Z3 ) ) @ ( sin @ real @ ( im @ Z3 ) ) ) ) ).
% Im_exp
thf(fact_3988_complex__eq,axiom,
! [A3: complex] :
( A3
= ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( re @ A3 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( im @ A3 ) ) ) ) ) ).
% complex_eq
thf(fact_3989_fun__complex__eq,axiom,
! [A: $tType,F3: A > complex] :
( F3
= ( ^ [X4: A] : ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( re @ ( F3 @ X4 ) ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( im @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% fun_complex_eq
thf(fact_3990_times__complex_Ocode,axiom,
( ( times_times @ complex )
= ( ^ [X4: complex,Y4: complex] : ( complex2 @ ( minus_minus @ real @ ( times_times @ real @ ( re @ X4 ) @ ( re @ Y4 ) ) @ ( times_times @ real @ ( im @ X4 ) @ ( im @ Y4 ) ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X4 ) @ ( im @ Y4 ) ) @ ( times_times @ real @ ( im @ X4 ) @ ( re @ Y4 ) ) ) ) ) ) ).
% times_complex.code
thf(fact_3991_complex__div__gt__0,axiom,
! [A3: complex,B2: complex] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A3 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) ) )
& ( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A3 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) ) ) ) ).
% complex_div_gt_0
thf(fact_3992_exp__eq__polar,axiom,
( ( exp @ complex )
= ( ^ [Z4: complex] : ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( exp @ real @ ( re @ Z4 ) ) ) @ ( cis @ ( im @ Z4 ) ) ) ) ) ).
% exp_eq_polar
thf(fact_3993_Im__power2,axiom,
! [X: complex] :
( ( im @ ( power_power @ complex @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% Im_power2
thf(fact_3994_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs2: list @ nat] :
( ( nat_list_encode @ ( cons @ nat @ X @ Xs2 ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_3995_series__comparison__complex,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G3: nat > complex,N7: nat,F3: nat > A] :
( ( summable @ complex @ G3 )
=> ( ! [N3: nat] : ( member @ complex @ ( G3 @ N3 ) @ ( real_Vector_Reals @ complex ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( G3 @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( real_V7770717601297561774m_norm @ complex @ ( G3 @ N3 ) ) ) )
=> ( summable @ A @ F3 ) ) ) ) ) ) ).
% series_comparison_complex
thf(fact_3996_complex__diff__cnj,axiom,
! [Z3: complex] :
( ( minus_minus @ complex @ Z3 @ ( cnj @ Z3 ) )
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( im @ Z3 ) ) ) @ imaginary_unit ) ) ).
% complex_diff_cnj
thf(fact_3997_Re__divide,axiom,
! [X: complex,Y: complex] :
( ( re @ ( divide_divide @ complex @ X @ Y ) )
= ( divide_divide @ real @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times @ real @ ( im @ X ) @ ( 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_3998_complex__mult__cnj,axiom,
! [Z3: complex] :
( ( times_times @ complex @ Z3 @ ( cnj @ Z3 ) )
= ( real_Vector_of_real @ complex @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_mult_cnj
thf(fact_3999_Im__divide,axiom,
! [X: complex,Y: complex] :
( ( im @ ( divide_divide @ complex @ X @ Y ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( times_times @ real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times @ real @ ( re @ X ) @ ( 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_4000_complex__abs__le__norm,axiom,
! [Z3: complex] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( abs_abs @ real @ ( re @ Z3 ) ) @ ( abs_abs @ real @ ( im @ Z3 ) ) ) @ ( times_times @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) ) ) ).
% complex_abs_le_norm
thf(fact_4001_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_4002_csqrt_Osimps_I2_J,axiom,
! [Z3: complex] :
( ( im @ ( csqrt @ Z3 ) )
= ( times_times @ real
@ ( if @ real
@ ( ( im @ Z3 )
= ( zero_zero @ real ) )
@ ( one_one @ real )
@ ( sgn_sgn @ real @ ( im @ Z3 ) ) )
@ ( sqrt @ ( divide_divide @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% csqrt.simps(2)
thf(fact_4003_csqrt__of__real__nonpos,axiom,
! [X: complex] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( re @ X ) @ ( zero_zero @ real ) )
=> ( ( csqrt @ X )
= ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sqrt @ ( abs_abs @ real @ ( re @ X ) ) ) ) ) ) ) ) ).
% csqrt_of_real_nonpos
thf(fact_4004_csqrt__minus,axiom,
! [X: complex] :
( ( ( ord_less @ real @ ( im @ X ) @ ( zero_zero @ real ) )
| ( ( ( im @ X )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ X ) ) ) )
=> ( ( csqrt @ ( uminus_uminus @ complex @ X ) )
= ( times_times @ complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% csqrt_minus
thf(fact_4005_rec__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F3: nat > A > A,V3: num,N: nat] :
( ( rec_nat @ A @ A3 @ F3 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V3 ) @ N ) )
= ( F3 @ ( plus_plus @ nat @ ( pred_numeral @ V3 ) @ N ) @ ( rec_nat @ A @ A3 @ F3 @ ( plus_plus @ nat @ ( pred_numeral @ V3 ) @ N ) ) ) ) ).
% rec_nat_add_eq_if
thf(fact_4006_case__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F3: nat > A,V3: num,N: nat] :
( ( case_nat @ A @ A3 @ F3 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V3 ) @ N ) )
= ( F3 @ ( plus_plus @ nat @ ( pred_numeral @ V3 ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_4007_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N5: nat,A7: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A7 ) @ N5 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A7 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N5 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A7 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_4008_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_4009_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se4730199178511100633sh_bit @ A @ N @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_4010_push__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% push_bit_of_0
thf(fact_4011_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_4012_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_4013_case__nat__numeral,axiom,
! [A: $tType,A3: A,F3: nat > A,V3: num] :
( ( case_nat @ A @ A3 @ F3 @ ( numeral_numeral @ nat @ V3 ) )
= ( F3 @ ( pred_numeral @ V3 ) ) ) ).
% case_nat_numeral
thf(fact_4014_rec__nat__numeral,axiom,
! [A: $tType,A3: A,F3: nat > A > A,V3: num] :
( ( rec_nat @ A @ A3 @ F3 @ ( numeral_numeral @ nat @ V3 ) )
= ( F3 @ ( pred_numeral @ V3 ) @ ( rec_nat @ A @ A3 @ F3 @ ( pred_numeral @ V3 ) ) ) ) ).
% rec_nat_numeral
thf(fact_4015_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_4016_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_4017_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_4018_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_4019_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_4020_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_4021_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_4022_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_4023_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_4024_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_4025_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_4026_rec__nat__0__imp,axiom,
! [A: $tType,F3: nat > A,F1: A,F22: nat > A > A] :
( ( F3
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F3 @ ( zero_zero @ nat ) )
= F1 ) ) ).
% rec_nat_0_imp
thf(fact_4027_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_4028_bit__push__bit__iff__nat,axiom,
! [M2: nat,Q2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ Q2 ) @ N )
= ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ Q2 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_4029_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_4030_max__Suc2,axiom,
! [M2: nat,N: nat] :
( ( ord_max @ nat @ M2 @ ( suc @ N ) )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M7: nat] : ( suc @ ( ord_max @ nat @ M7 @ N ) )
@ M2 ) ) ).
% max_Suc2
thf(fact_4031_max__Suc1,axiom,
! [N: nat,M2: nat] :
( ( ord_max @ nat @ ( suc @ N ) @ M2 )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M7: nat] : ( suc @ ( ord_max @ nat @ N @ M7 ) )
@ M2 ) ) ).
% max_Suc1
thf(fact_4032_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_4033_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A7: A,N5: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A7 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_4034_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_4035_push__bit__nat__def,axiom,
( ( bit_se4730199178511100633sh_bit @ nat )
= ( ^ [N5: nat,M6: nat] : ( times_times @ nat @ M6 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ).
% push_bit_nat_def
thf(fact_4036_push__bit__int__def,axiom,
( ( bit_se4730199178511100633sh_bit @ int )
= ( ^ [N5: nat,K3: int] : ( times_times @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ).
% push_bit_int_def
thf(fact_4037_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N5: nat,A7: A] : ( times_times @ A @ A7 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_4038_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N5: nat,A7: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A7 @ K3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% take_bit_sum
thf(fact_4039_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_4040_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X4: A,F4: nat > A,N5: nat] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ X4
@ ( F4 @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_4041_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P2: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P2 @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ~ ( ( ( Nat
= ( zero_zero @ nat ) )
& ~ ( P2 @ F1 ) )
| ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
& ~ ( P2 @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_4042_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P2: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P2 @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ( ( Nat
= ( zero_zero @ nat ) )
=> ( P2 @ F1 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P2 @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_4043_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X24: nat] : X24 ) ) ).
% pred_def
thf(fact_4044_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_4045_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_4046_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N5: nat,A7: A] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ A7
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_4047_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,J: code_integer] :
( if @ nat
@ ( J
= ( 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_4048_drop__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% drop_bit_of_0
thf(fact_4049_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_4050_drop__bit__of__bool,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,B2: $o] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( zero_neq_one_of_bool @ A @ B2 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( N
= ( zero_zero @ nat ) )
& B2 ) ) ) ) ).
% drop_bit_of_bool
thf(fact_4051_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_4052_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_4053_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_4054_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_4055_drop__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% drop_bit_of_1
thf(fact_4056_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_4057_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_4058_fst__divmod__nat,axiom,
! [M2: nat,N: nat] :
( ( product_fst @ nat @ nat @ ( divmod_nat @ M2 @ N ) )
= ( divide_divide @ nat @ M2 @ N ) ) ).
% fst_divmod_nat
thf(fact_4059_nat__of__integer__non__positive,axiom,
! [K2: code_integer] :
( ( ord_less_eq @ code_integer @ K2 @ ( zero_zero @ code_integer ) )
=> ( ( code_nat_of_integer @ K2 )
= ( zero_zero @ nat ) ) ) ).
% nat_of_integer_non_positive
thf(fact_4060_snd__divmod__nat,axiom,
! [M2: nat,N: nat] :
( ( product_snd @ nat @ nat @ ( divmod_nat @ M2 @ N ) )
= ( modulo_modulo @ nat @ M2 @ N ) ) ).
% snd_divmod_nat
thf(fact_4061_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_4062_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_4063_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_4064_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C4: A > B > C,P5: product_prod @ A @ B] : ( C4 @ ( product_fst @ A @ B @ P5 ) @ ( product_snd @ A @ B @ P5 ) ) ) ) ).
% case_prod_unfold
thf(fact_4065_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_4066_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_4067_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X16: A,X24: B] : X24 ) ) ).
% snd_def
thf(fact_4068_The__case__prod,axiom,
! [B: $tType,A: $tType,P2: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy2: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Xy2 ) @ ( product_snd @ A @ B @ Xy2 ) ) ) ) ).
% The_case_prod
thf(fact_4069_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X16: A,X24: B] : X16 ) ) ).
% fst_def
thf(fact_4070_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y5: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y5 = Z2 )
= ( ^ [S6: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S6 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S6 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_4071_prod__eqI,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P = Q2 ) ) ) ).
% prod_eqI
thf(fact_4072_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_4073_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_4074_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A3 )
=> ( Y = A3 ) ) ).
% snd_eqD
thf(fact_4075_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_4076_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A3 )
=> ( X = A3 ) ) ).
% fst_eqD
thf(fact_4077_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_4078_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_4079_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,X: A,Y: B,A3: product_prod @ A @ B] :
( ( P2 @ X @ Y )
=> ( ( A3
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P2 @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_4080_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
=> ( A6 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_4081_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_4082_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_4083_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P2 @ ( F3 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_4084_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P2 @ ( F3 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_4085_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= A3 )
= ( ( bit_se4197421643247451524op_bit @ A @ N @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_4086_in__set__zip,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Xs2: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys ) ) )
= ( ? [N5: nat] :
( ( ( nth @ A @ Xs2 @ N5 )
= ( product_fst @ A @ B @ P ) )
& ( ( nth @ B @ Ys @ N5 )
= ( product_snd @ A @ B @ P ) )
& ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ord_less @ nat @ N5 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ).
% in_set_zip
thf(fact_4087_divides__aux__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique5940410009612947441es_aux @ A )
= ( ^ [Qr: product_prod @ A @ A] :
( ( product_snd @ A @ A @ Qr )
= ( zero_zero @ A ) ) ) ) ) ).
% divides_aux_def
thf(fact_4088_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_4089_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_4090_in__set__enumerate__eq,axiom,
! [A: $tType,P: product_prod @ nat @ A,N: nat,Xs2: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs2 ) ) )
= ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) )
& ( ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P ) @ N ) )
= ( product_snd @ nat @ A @ P ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_4091_nat__of__integer__code__post_I1_J,axiom,
( ( code_nat_of_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ nat ) ) ).
% nat_of_integer_code_post(1)
thf(fact_4092_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_4093_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_4094_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,J: code_integer] :
( if @ int
@ ( J
= ( 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_4095_conjI__realizer,axiom,
! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_4096_exI__realizer,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Y: A,X: B] :
( ( P2 @ Y @ X )
=> ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_4097_times__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( times_times @ code_integer @ X @ Xa2 ) )
= ( times_times @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% times_integer.rep_eq
thf(fact_4098_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_4099_bezw_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_4100_bezw_Osimps,axiom,
( bezw
= ( ^ [X4: nat,Y4: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y4
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y4 @ ( modulo_modulo @ nat @ X4 @ Y4 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y4 @ ( modulo_modulo @ nat @ X4 @ Y4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y4 @ ( modulo_modulo @ nat @ X4 @ Y4 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X4 @ Y4 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_4101_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_4102_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_4103_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_4104_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_4105_Divides_Oadjust__mod__def,axiom,
( adjust_mod
= ( ^ [L2: int,R: int] :
( if @ int
@ ( R
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ L2 @ R ) ) ) ) ).
% Divides.adjust_mod_def
thf(fact_4106_bezw_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa2 ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) ) ) ) ) ).
% bezw.pelims
thf(fact_4107_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_4108_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R3 ) )
= ( ( ( 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
@ ^ [Y4: A,Z4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y4 @ Z4 ) @ R3 )
@ X4 ) ) ) ) ).
% listrel_iff_zip
thf(fact_4109_semiring__char__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiri4206861660011772517g_char @ A )
= ( ^ [Uu4: itself @ A] :
( gcd_Gcd @ nat
@ ( collect @ nat
@ ^ [N5: nat] :
( ( semiring_1_of_nat @ A @ N5 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% semiring_char_def
thf(fact_4110_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I2 )
& ( ord_less @ A @ I2 @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_4111_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% finite_greaterThanLessThan_int
thf(fact_4112_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_4113_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_4114_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_4115_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ioo_iff
thf(fact_4116_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ioo
thf(fact_4117_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_4118_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_4119_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_4120_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_4121_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_4122_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_4123_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_4124_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_4125_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_4126_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_4127_ivl__disj__un__one_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_4128_list__eq__iff__zip__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z2: list @ A] : Y5 = Z2 )
= ( ^ [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,Z2: A] : Y5 = Z2
@ X4 ) ) ) ) ) ).
% list_eq_iff_zip_eq
thf(fact_4129_concat__eq__concat__iff,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ! [X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X3 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y4: list @ A,Z4: list @ A] :
( ( size_size @ ( list @ A ) @ Y4 )
= ( size_size @ ( list @ A ) @ Z4 ) )
@ X3 ) )
=> ( ( ( 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_4130_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 ) )
=> ( ! [X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X3 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y4: list @ A,Z4: list @ A] :
( ( size_size @ ( list @ A ) @ Y4 )
= ( size_size @ ( list @ A ) @ Z4 ) )
@ X3 ) )
=> ( Xs2 = Ys ) ) ) ) ).
% concat_injective
thf(fact_4131_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_4132_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_4133_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_4134_bounded__linear__axioms_Ointro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ? [K8: real] :
! [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X3 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K8 ) )
=> ( real_V4916620083959148203axioms @ A @ B @ F3 ) ) ) ).
% bounded_linear_axioms.intro
thf(fact_4135_bounded__linear__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( real_V4916620083959148203axioms @ A @ B )
= ( ^ [F4: A > B] :
? [K6: real] :
! [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F4 @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ K6 ) ) ) ) ) ).
% bounded_linear_axioms_def
thf(fact_4136_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A7: A,Xs: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( compow @ ( A > A ) @ N5 @ ( times_times @ A @ A7 ) @ ( F4 @ ( nth @ B @ Xs @ N5 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_4137_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_4138_Suc__funpow,axiom,
! [N: nat] :
( ( compow @ ( nat > nat ) @ N @ suc )
= ( plus_plus @ nat @ N ) ) ).
% Suc_funpow
thf(fact_4139_funpow__0,axiom,
! [A: $tType,F3: A > A,X: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F3 @ X )
= X ) ).
% funpow_0
thf(fact_4140_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_4141_funpow__swap1,axiom,
! [A: $tType,F3: A > A,N: nat,X: A] :
( ( F3 @ ( compow @ ( A > A ) @ N @ F3 @ X ) )
= ( compow @ ( A > A ) @ N @ F3 @ ( F3 @ X ) ) ) ).
% funpow_swap1
thf(fact_4142_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_4143_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_4144_funpow__times__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [F3: A > nat,X: A] :
( ( compow @ ( A > A ) @ ( F3 @ X ) @ ( times_times @ A @ X ) )
= ( times_times @ A @ ( power_power @ A @ X @ ( F3 @ X ) ) ) ) ) ).
% funpow_times_power
thf(fact_4145_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_4146_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_4147_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N5: nat] : ( compow @ ( A > A ) @ N5 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_4148_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_4149_relpowp__bot,axiom,
! [A: $tType,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( compow @ ( A > A > $o ) @ N @ ( bot_bot @ ( A > A > $o ) ) )
= ( bot_bot @ ( A > A > $o ) ) ) ) ).
% relpowp_bot
thf(fact_4150_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N5: nat,P4: A > A > $o,X4: A,Y4: A] :
? [F4: nat > A] :
( ( ( F4 @ ( zero_zero @ nat ) )
= X4 )
& ( ( F4 @ N5 )
= Y4 )
& ! [I: nat] :
( ( ord_less @ nat @ I @ N5 )
=> ( P4 @ ( F4 @ I ) @ ( F4 @ ( suc @ I ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_4151_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_4152_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_4153_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_4154_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_4155_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_4156_sub__num__simps_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K2 ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K2 @ L ) ) ) ) ).
% sub_num_simps(9)
thf(fact_4157_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_4158_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_4159_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_4160_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_4161_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_4162_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_4163_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_4164_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_4165_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_4166_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_4167_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_4168_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_4169_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_4170_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_4171_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_4172_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_4173_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_4174_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_4175_relpowp__Suc__E,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P2 @ X @ Z3 )
=> ~ ! [Y3: A] :
( ( compow @ ( A > A > $o ) @ N @ P2 @ X @ Y3 )
=> ~ ( P2 @ Y3 @ Z3 ) ) ) ).
% relpowp_Suc_E
thf(fact_4176_relpowp__Suc__I,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Y: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ N @ P2 @ X @ Y )
=> ( ( P2 @ Y @ Z3 )
=> ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P2 @ X @ Z3 ) ) ) ).
% relpowp_Suc_I
thf(fact_4177_relpowp__Suc__D2,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P2 @ X @ Z3 )
=> ? [Y3: A] :
( ( P2 @ X @ Y3 )
& ( compow @ ( A > A > $o ) @ N @ P2 @ Y3 @ Z3 ) ) ) ).
% relpowp_Suc_D2
thf(fact_4178_relpowp__Suc__E2,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P2 @ X @ Z3 )
=> ~ ! [Y3: A] :
( ( P2 @ X @ Y3 )
=> ~ ( compow @ ( A > A > $o ) @ N @ P2 @ Y3 @ Z3 ) ) ) ).
% relpowp_Suc_E2
thf(fact_4179_relpowp__Suc__I2,axiom,
! [A: $tType,P2: A > A > $o,X: A,Y: A,N: nat,Z3: A] :
( ( P2 @ X @ Y )
=> ( ( compow @ ( A > A > $o ) @ N @ P2 @ Y @ Z3 )
=> ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P2 @ X @ Z3 ) ) ) ).
% relpowp_Suc_I2
thf(fact_4180_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R2: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R2 )
= ( ^ [Y5: A,Z2: A] : Y5 = Z2 ) ) ).
% relpowp.simps(1)
thf(fact_4181_relpowp__0__E,axiom,
! [A: $tType,P2: A > A > $o,X: A,Y: A] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P2 @ X @ Y )
=> ( X = Y ) ) ).
% relpowp_0_E
thf(fact_4182_relpowp__0__I,axiom,
! [A: $tType,P2: A > A > $o,X: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P2 @ X @ X ) ).
% relpowp_0_I
thf(fact_4183_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_4184_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_4185_sub__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M2 ) )
= ( ord_less @ num @ M2 @ N ) ) ) ).
% sub_positive
thf(fact_4186_sub__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M2: num] :
( ( ord_less @ A @ ( neg_numeral_sub @ A @ N @ M2 ) @ ( zero_zero @ A ) )
= ( ord_less @ num @ N @ M2 ) ) ) ).
% sub_negative
thf(fact_4187_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_4188_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_4189_relpowp__E2,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ N @ P2 @ X @ Z3 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z3 ) )
=> ~ ! [Y3: A,M: nat] :
( ( N
= ( suc @ M ) )
=> ( ( P2 @ X @ Y3 )
=> ~ ( compow @ ( A > A > $o ) @ M @ P2 @ Y3 @ Z3 ) ) ) ) ) ).
% relpowp_E2
thf(fact_4190_relpowp__E,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ N @ P2 @ X @ Z3 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z3 ) )
=> ~ ! [Y3: A,M: nat] :
( ( N
= ( suc @ M ) )
=> ( ( compow @ ( A > A > $o ) @ M @ P2 @ X @ Y3 )
=> ~ ( P2 @ Y3 @ Z3 ) ) ) ) ) ).
% relpowp_E
thf(fact_4191_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_4192_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_4193_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X4: nat,Y4: nat] : ( ord_less_eq @ nat @ Y4 @ X4 )
@ ^ [X4: nat,Y4: nat] : ( ord_less @ nat @ Y4 @ X4 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_4194_bit_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( boolea2506097494486148201lgebra @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.abstract_boolean_algebra_axioms
thf(fact_4195_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_4196_real__root__eq__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_4197_root__0,axiom,
! [X: real] :
( ( root @ ( zero_zero @ nat ) @ X )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_4198_real__root__eq__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% real_root_eq_0_iff
thf(fact_4199_real__root__less__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_4200_real__root__le__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_4201_real__root__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( one_one @ real ) )
= ( one_one @ real ) ) ) ).
% real_root_one
thf(fact_4202_real__root__eq__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% real_root_eq_1_iff
thf(fact_4203_real__root__gt__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_4204_real__root__lt__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_lt_0_iff
thf(fact_4205_real__root__ge__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_4206_real__root__le__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_le_0_iff
thf(fact_4207_real__root__gt__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_4208_real__root__lt__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_lt_1_iff
thf(fact_4209_real__root__ge__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_4210_real__root__le__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_le_1_iff
thf(fact_4211_real__root__pow__pos2,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_4212_real__root__mult__exp,axiom,
! [M2: nat,N: nat,X: real] :
( ( root @ ( times_times @ nat @ M2 @ N ) @ X )
= ( root @ M2 @ ( root @ N @ X ) ) ) ).
% real_root_mult_exp
thf(fact_4213_real__root__mult,axiom,
! [N: nat,X: real,Y: real] :
( ( root @ N @ ( times_times @ real @ X @ Y ) )
= ( times_times @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).
% real_root_mult
thf(fact_4214_real__root__less__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_4215_real__root__le__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_4216_real__root__power,axiom,
! [N: nat,X: real,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X @ K2 ) )
= ( power_power @ real @ ( root @ N @ X ) @ K2 ) ) ) ).
% real_root_power
thf(fact_4217_real__root__abs,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( abs_abs @ real @ X ) )
= ( abs_abs @ real @ ( root @ N @ X ) ) ) ) ).
% real_root_abs
thf(fact_4218_sgn__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( sgn_sgn @ real @ ( root @ N @ X ) )
= ( sgn_sgn @ real @ X ) ) ) ).
% sgn_root
thf(fact_4219_real__root__gt__zero,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_4220_real__root__strict__decreasing,axiom,
! [N: nat,N7: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N7 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( root @ N7 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_4221_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_4222_real__root__pos__pos,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_4223_real__root__strict__increasing,axiom,
! [N: nat,N7: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N7 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N7 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_4224_real__root__decreasing,axiom,
! [N: nat,N7: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( root @ N7 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_4225_real__root__pow__pos,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_4226_real__root__pos__unique,axiom,
! [N: nat,Y: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( power_power @ real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_4227_real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N @ ( power_power @ real @ X @ N ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_4228_real__root__increasing,axiom,
! [N: nat,N7: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N7 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_4229_sgn__power__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N @ X ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N @ X ) ) @ N ) )
= X ) ) ).
% sgn_power_root
thf(fact_4230_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_4231_ln__root,axiom,
! [N: nat,B2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ln_ln @ real @ ( root @ N @ B2 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% ln_root
thf(fact_4232_log__root,axiom,
! [N: nat,A3: real,B2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( log2 @ B2 @ ( root @ N @ A3 ) )
= ( divide_divide @ real @ ( log2 @ B2 @ A3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_root
thf(fact_4233_log__base__root,axiom,
! [N: nat,B2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( log2 @ ( root @ N @ B2 ) @ X )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log2 @ B2 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_4234_split__root,axiom,
! [P2: real > $o,N: nat,X: real] :
( ( P2 @ ( root @ N @ X ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P2 @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ! [Y4: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y4 ) @ ( power_power @ real @ ( abs_abs @ real @ Y4 ) @ N ) )
= X )
=> ( P2 @ Y4 ) ) ) ) ) ).
% split_root
thf(fact_4235_root__powr__inverse,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N @ X )
= ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_4236_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P ) )
= ( ? [A7: B] :
( P
= ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_4237_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z3: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z3 ) )
=> ( ( product_snd @ A @ B @ X )
= Z3 ) ) ).
% sndI
thf(fact_4238_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P: product_prod @ A @ B] :
( ( A3
= ( product_fst @ A @ B @ P ) )
= ( ? [B5: B] :
( P
= ( product_Pair @ A @ B @ A3 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_4239_powr__eq__0__iff,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [W2: A,Z3: A] :
( ( ( powr @ A @ W2 @ Z3 )
= ( zero_zero @ A ) )
= ( W2
= ( zero_zero @ A ) ) ) ) ).
% powr_eq_0_iff
thf(fact_4240_powr__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [Z3: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z3 )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_4241_powr__zero__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% powr_zero_eq_one
thf(fact_4242_powr__powr,axiom,
! [X: real,A3: real,B2: real] :
( ( powr @ real @ ( powr @ real @ X @ A3 ) @ B2 )
= ( powr @ real @ X @ ( times_times @ real @ A3 @ B2 ) ) ) ).
% powr_powr
thf(fact_4243_powr__mult,axiom,
! [X: real,Y: real,A3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( times_times @ real @ X @ Y ) @ A3 )
= ( times_times @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y @ A3 ) ) ) ) ) ).
% powr_mult
thf(fact_4244_divide__powr__uminus,axiom,
! [A3: real,B2: real,C3: real] :
( ( divide_divide @ real @ A3 @ ( powr @ real @ B2 @ C3 ) )
= ( times_times @ real @ A3 @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ C3 ) ) ) ) ).
% divide_powr_uminus
thf(fact_4245_ln__powr,axiom,
! [X: real,Y: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ ( powr @ real @ X @ Y ) )
= ( times_times @ real @ Y @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_powr
thf(fact_4246_log__powr,axiom,
! [X: real,B2: real,Y: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( log2 @ B2 @ ( powr @ real @ X @ Y ) )
= ( times_times @ real @ Y @ ( log2 @ B2 @ X ) ) ) ) ).
% log_powr
thf(fact_4247_powr__add,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X: A,A3: A,B2: A] :
( ( powr @ A @ X @ ( plus_plus @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( powr @ A @ X @ A3 ) @ ( powr @ A @ X @ B2 ) ) ) ) ).
% powr_add
thf(fact_4248_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B] :
( ( size_size @ ( product_prod @ A @ B ) @ X )
!= ( zero_zero @ nat ) ) ).
% prod.size_neq
thf(fact_4249_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X: sum_sum @ A @ B] :
( ( size_size @ ( sum_sum @ A @ B ) @ X )
!= ( zero_zero @ nat ) ) ).
% sum.size_neq
thf(fact_4250_powr__mult__base,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( times_times @ real @ X @ ( powr @ real @ X @ Y ) )
= ( powr @ real @ X @ ( plus_plus @ real @ ( one_one @ real ) @ Y ) ) ) ) ).
% powr_mult_base
thf(fact_4251_ln__powr__bound2,axiom,
! [X: real,A3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X ) @ A3 ) @ ( times_times @ real @ ( powr @ real @ A3 @ A3 ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_4252_log__add__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ ( log2 @ B2 @ X ) @ Y )
= ( log2 @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ Y ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_4253_add__log__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ Y @ ( log2 @ B2 @ X ) )
= ( log2 @ B2 @ ( times_times @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_4254_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X4: A,A7: A] :
( if @ A
@ ( X4
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A7 @ ( ln_ln @ A @ X4 ) ) ) ) ) ) ) ).
% powr_def
thf(fact_4255_log__minus__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ ( log2 @ B2 @ X ) @ Y )
= ( log2 @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_4256_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z3: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z3 ) )
=> ( ( product_fst @ A @ B @ X )
= Y ) ) ).
% fstI
thf(fact_4257_times__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y4 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V4 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) )
@ Xa2
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_4258_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_4259_eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ one2 ) ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_4260_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_4261_num__of__nat__numeral__eq,axiom,
! [Q2: num] :
( ( num_of_nat @ ( numeral_numeral @ nat @ Q2 ) )
= Q2 ) ).
% num_of_nat_numeral_eq
thf(fact_4262_iszero__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: num] :
( ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% iszero_neg_numeral
thf(fact_4263_eq__Abs__Integ,axiom,
! [Z3: int] :
~ ! [X3: nat,Y3: nat] :
( Z3
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X3 @ Y3 ) ) ) ).
% eq_Abs_Integ
thf(fact_4264_iszero__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ring_1_iszero @ A @ ( zero_zero @ A ) ) ) ).
% iszero_0
thf(fact_4265_iszero__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_iszero @ A )
= ( ^ [Z4: A] :
( Z4
= ( zero_zero @ A ) ) ) ) ) ).
% iszero_def
thf(fact_4266_not__iszero__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W2: num] :
~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W2 ) ) ) ).
% not_iszero_numeral
thf(fact_4267_not__iszero__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% not_iszero_1
thf(fact_4268_eq__iff__iszero__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [X4: A,Y4: A] : ( ring_1_iszero @ A @ ( minus_minus @ A @ X4 @ Y4 ) ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_4269_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_4270_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_4271_eq__numeral__iff__iszero_I9_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( numeral_numeral @ A @ X )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_4272_not__iszero__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ one2 ) ) ) ).
% not_iszero_Numeral1
thf(fact_4273_not__iszero__neg__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_iszero_neg_1
thf(fact_4274_eq__numeral__iff__iszero_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( numeral_numeral @ A @ X )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_4275_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_4276_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N5: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N5 @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_4277_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_4278_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_4279_eq__numeral__iff__iszero_I11_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_4280_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_4281_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_4282_eq__numeral__iff__iszero_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_4283_eq__numeral__iff__iszero_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( numeral_numeral @ A @ X )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_4284_eq__numeral__iff__iszero_I4_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ Y @ X ) ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_4285_uminus__int_Oabs__eq,axiom,
! [X: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X4 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_4286_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_4287_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_4288_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_4289_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_4290_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) )
@ Xa2
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_4291_eq__numeral__iff__iszero_I5_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( numeral_numeral @ A @ X )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ one2 ) ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_4292_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_4293_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y4 @ V4 ) ) )
@ Xa2
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_4294_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) )
@ Xa2
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_4295_listrel1__iff__update,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R3 ) )
= ( ? [Y4: A,N5: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs2 @ N5 ) @ Y4 ) @ R3 )
& ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( Ys
= ( list_update @ A @ Xs2 @ N5 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_4296_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,P: B > A,I2: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( P @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I2 @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P @ ( insert @ B @ I2 @ I5 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P @ I5 ) ) )
& ( ~ ( member @ B @ I2 @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P @ ( insert @ B @ I2 @ I5 ) )
= ( times_times @ A @ ( P @ I2 ) @ ( groups1962203154675924110t_prod @ B @ A @ P @ I5 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_4297_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_4298_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_4299_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_4300_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs2: list @ A,Y: A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R3 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
& ( Xs2 = Ys ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_4301_listrel1__eq__len,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R3 ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_4302_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I: B] : ( times_times @ A @ ( G3 @ I ) @ ( H @ I ) )
@ I5 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G3 @ I5 ) @ ( groups1962203154675924110t_prod @ B @ A @ H @ I5 ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_4303_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y @ Xs2 ) ) @ ( listrel1 @ A @ R3 ) ) ) ).
% listrel1I1
thf(fact_4304_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs2: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Ys ) @ ( listrel1 @ A @ R3 ) )
=> ( ! [Y3: A] :
( ( Ys
= ( cons @ A @ Y3 @ Xs2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R3 ) )
=> ~ ! [Zs2: list @ A] :
( ( Ys
= ( cons @ A @ X @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Zs2 ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_4305_Cons__listrel1E2,axiom,
! [A: $tType,Xs2: list @ A,Y: A,Ys: list @ A,R3: 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 @ R3 ) )
=> ( ! [X3: A] :
( ( Xs2
= ( cons @ A @ X3 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R3 ) )
=> ~ ! [Zs2: list @ A] :
( ( Xs2
= ( cons @ A @ Y @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs2 @ Ys ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_4306_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( G3 @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( H @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I: B] : ( times_times @ A @ ( G3 @ I ) @ ( H @ I ) )
@ I5 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G3 @ I5 ) @ ( groups1962203154675924110t_prod @ B @ A @ H @ I5 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_4307_listrel1E,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R3 ) )
=> ~ ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ! [Us2: list @ A,Vs2: list @ A] :
( ( Xs2
= ( append @ A @ Us2 @ ( cons @ A @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append @ A @ Us2 @ ( cons @ A @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_4308_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R3: 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 @ X @ Y ) @ R3 )
=> ( ( Xs2
= ( append @ A @ Us @ ( cons @ A @ X @ 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 @ R3 ) ) ) ) ) ).
% listrel1I
thf(fact_4309_sorted__list__of__set__greaterThanLessThan,axiom,
! [I2: nat,J2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ J2 )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I2 @ J2 ) )
= ( cons @ nat @ ( suc @ I2 ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I2 ) @ J2 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_4310_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs2: list @ A,X: A,Ys: list @ A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R3 ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( listrel1 @ A @ R3 ) )
& ( X = Y ) )
| ( ( Xs2 = Ys )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_4311_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J2: nat,I2: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J2 @ ( suc @ I2 ) ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I2 @ J2 ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_4312_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X4: int,Xa3: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y4: nat,Z4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y4 @ V4 ) @ ( plus_plus @ nat @ U2 @ Z4 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa3 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_4313_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R: 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,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R ) ) ) ) ) ) ).
% listrel_def
thf(fact_4314_nths__Cons,axiom,
! [A: $tType,X: A,L: list @ A,A6: set @ nat] :
( ( nths @ A @ ( cons @ A @ X @ L ) @ A6 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A6 ) @ ( cons @ A @ X @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J: nat] : ( member @ nat @ ( suc @ J ) @ A6 ) ) ) ) ) ).
% nths_Cons
thf(fact_4315_image__minus__const__atLeastLessThan__nat,axiom,
! [C3: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C3 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I: nat] : ( minus_minus @ nat @ I @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C3 ) @ ( minus_minus @ nat @ Y @ C3 ) ) ) )
& ( ~ ( ord_less @ nat @ C3 @ Y )
=> ( ( ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I: nat] : ( minus_minus @ nat @ I @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I: nat] : ( minus_minus @ nat @ I @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_4316_bij__betw__Suc,axiom,
! [M5: set @ nat,N7: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M5 @ N7 )
= ( ( image2 @ nat @ nat @ suc @ M5 )
= N7 ) ) ).
% bij_betw_Suc
thf(fact_4317_image__add__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A,J2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K2 ) @ ( set_or1337092689740270186AtMost @ A @ I2 @ J2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J2 @ K2 ) ) ) ) ).
% image_add_atLeastAtMost
thf(fact_4318_image__diff__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ D3 @ B2 ) @ ( minus_minus @ A @ D3 @ A3 ) ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_4319_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_4320_image__add__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A,J2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K2 ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ J2 ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J2 @ K2 ) ) ) ) ).
% image_add_atLeastLessThan
thf(fact_4321_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_4322_image__uminus__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= ( set_or5935395276787703475ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanLessThan
thf(fact_4323_image__Suc__atLeastAtMost,axiom,
! [I2: nat,J2: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I2 @ J2 ) )
= ( set_or1337092689740270186AtMost @ nat @ ( suc @ I2 ) @ ( suc @ J2 ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_4324_image__Suc__atLeastLessThan,axiom,
! [I2: nat,J2: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I2 @ J2 ) )
= ( set_or7035219750837199246ssThan @ nat @ ( suc @ I2 ) @ ( suc @ J2 ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_4325_bij__betw__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N7: set @ nat,A6: set @ A] :
( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N7 @ A6 )
= ( ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ N7 )
= A6 ) ) ) ).
% bij_betw_of_nat
thf(fact_4326_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A,J2: A] :
( ( image2 @ A @ A
@ ^ [N5: A] : ( plus_plus @ A @ N5 @ K2 )
@ ( set_or1337092689740270186AtMost @ A @ I2 @ J2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J2 @ K2 ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_4327_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( image2 @ A @ A
@ ^ [T3: A] : ( minus_minus @ A @ T3 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A3 @ D3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_4328_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A,J2: A] :
( ( image2 @ A @ A
@ ^ [N5: A] : ( plus_plus @ A @ N5 @ K2 )
@ ( set_or7035219750837199246ssThan @ A @ I2 @ J2 ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J2 @ K2 ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_4329_nths__singleton,axiom,
! [A: $tType,A6: set @ nat,X: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A6 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A6 )
= ( cons @ A @ X @ ( nil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A6 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A6 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_4330_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D3 @ A3 ) @ ( times_times @ A @ D3 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_4331_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A
@ ^ [C4: A] : ( divide_divide @ A @ C4 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D3 ) @ ( divide_divide @ A @ B2 @ D3 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_4332_zero__notin__Suc__image,axiom,
! [A6: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ A6 ) ) ).
% zero_notin_Suc_image
thf(fact_4333_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_4334_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_4335_surj__card__le,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image2 @ A @ B @ F3 @ A6 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B6 ) @ ( finite_card @ A @ A6 ) ) ) ) ).
% surj_card_le
thf(fact_4336_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_4337_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_4338_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_4339_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_4340_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_4341_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_4342_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
@ ^ [J: nat] : ( member @ nat @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ L ) ) @ A6 ) ) ) ) ) ).
% nths_append
thf(fact_4343_length__nths,axiom,
! [A: $tType,Xs2: list @ A,I5: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs2 @ I5 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( member @ nat @ I @ I5 ) ) ) ) ) ).
% length_nths
thf(fact_4344_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ X ) @ ( times_times @ A @ C3 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ Y ) @ ( times_times @ A @ C3 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_4345_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,C3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C3 ) @ ( times_times @ A @ Y @ C3 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C3 ) @ ( times_times @ A @ X @ C3 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_4346_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_4347_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_4348_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_4349_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_4350_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R3 ) )
= ( ^ [X4: list @ A,Y4: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X4 @ Y4 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_4351_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S3: set @ A,R2: set @ B,G3: A > B,F3: B > C] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ R2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G3 @ S3 ) @ R2 )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X4: A] : ( F3 @ ( G3 @ X4 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y4: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S3 )
& ( ( G3 @ X4 )
= Y4 ) ) ) ) )
@ ( F3 @ Y4 ) )
@ R2 ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_4352_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_4353_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J2: nat,I2: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J2 @ I2 ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I2 @ J2 ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_4354_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,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X4 ) ) ) ) ).
% uminus_int_def
thf(fact_4355_rat__inverse__code,axiom,
! [P: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( A7
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A7 ) @ B5 ) @ ( abs_abs @ int @ A7 ) ) )
@ ( quotient_of @ P ) ) ) ).
% rat_inverse_code
thf(fact_4356_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_4357_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_4358_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I2 )
& ( ord_less_eq @ A @ I2 @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_4359_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_4360_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_4361_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_4362_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ioc_iff
thf(fact_4363_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_4364_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_greaterThanAtMost
thf(fact_4365_image__minus__const__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) )
= ( set_or7035219750837199246ssThan @ A @ ( minus_minus @ A @ C3 @ B2 ) @ ( minus_minus @ A @ C3 @ A3 ) ) ) ) ).
% image_minus_const_greaterThanAtMost
thf(fact_4366_image__diff__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( minus_minus @ A @ C3 @ B2 ) @ ( minus_minus @ A @ C3 @ A3 ) ) ) ) ).
% image_diff_atLeastLessThan
thf(fact_4367_image__uminus__atLeastLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= ( set_or3652927894154168847AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastLessThan
thf(fact_4368_image__uminus__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanAtMost
thf(fact_4369_divide__rat__def,axiom,
( ( divide_divide @ rat )
= ( ^ [Q5: rat,R: rat] : ( times_times @ rat @ Q5 @ ( inverse_inverse @ rat @ R ) ) ) ) ).
% divide_rat_def
thf(fact_4370_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_4371_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P5: rat,Q5: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A7: int,C4: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B5: int,D4: int] : ( ord_less_eq @ int @ ( times_times @ int @ A7 @ D4 ) @ ( times_times @ int @ C4 @ B5 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_eq_code
thf(fact_4372_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P5: rat,Q5: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A7: int,C4: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B5: int,D4: int] : ( ord_less @ int @ ( times_times @ int @ A7 @ D4 ) @ ( times_times @ int @ C4 @ B5 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_code
thf(fact_4373_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_4374_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_4375_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_4376_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ioc
thf(fact_4377_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_4378_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_4379_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_4380_image__int__atLeastAtMost,axiom,
! [A3: nat,B2: nat] :
( ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastAtMost
thf(fact_4381_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_4382_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_4383_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_4384_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_4385_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_4386_image__int__atLeastLessThan,axiom,
! [A3: nat,B2: nat] :
( ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) )
= ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastLessThan
thf(fact_4387_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_4388_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X8: set @ A,A6: set @ ( product_prod @ A @ B ),Y7: set @ B,P2: A > B > $o,Q: A > B > $o] :
( ( X8
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 ) )
=> ( ( Y7
= ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ! [Xa4: B] :
( ( member @ B @ Xa4 @ Y7 )
=> ( ( P2 @ X3 @ Xa4 )
=> ( Q @ X3 @ Xa4 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) ) )
=> ( 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_4389_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_4390_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_4391_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_4392_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_4393_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_4394_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A7: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A7 @ B5 ) @ ( insert @ A @ A7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_4395_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_4396_ivl__disj__un__two__touch_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_4397_sorted__list__of__set__greaterThanAtMost,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ ( suc @ I2 ) @ J2 )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I2 @ J2 ) )
= ( cons @ nat @ ( suc @ I2 ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I2 ) @ J2 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_4398_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image2 @ int @ int
@ ^ [X4: int] : ( plus_plus @ int @ X4 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_4399_ivl__disj__un__singleton_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_4400_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_4401_ivl__disj__un__singleton_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_4402_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
=> ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U )
= ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_4403_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y4 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V4 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_4404_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_4405_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y4 @ V4 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_4406_rat__minus__code,axiom,
! [P: rat,Q2: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A7 @ D4 ) @ ( times_times @ int @ B5 @ C4 ) ) @ ( times_times @ int @ C4 @ D4 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_minus_code
thf(fact_4407_finite__greaterThanAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% finite_greaterThanAtMost_int
thf(fact_4408_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_greaterThanAtMost_int
thf(fact_4409_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_4410_normalize__crossproduct,axiom,
! [Q2: int,S2: int,P: int,R3: int] :
( ( Q2
!= ( zero_zero @ int ) )
=> ( ( S2
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P @ Q2 ) )
= ( normalize @ ( product_Pair @ int @ int @ R3 @ S2 ) ) )
=> ( ( times_times @ int @ P @ S2 )
= ( times_times @ int @ R3 @ Q2 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_4411_rat__times__code,axiom,
! [P: rat,Q2: rat] :
( ( quotient_of @ ( times_times @ rat @ P @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A7 @ B5 ) @ ( times_times @ int @ C4 @ D4 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_times_code
thf(fact_4412_rat__divide__code,axiom,
! [P: rat,Q2: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A7 @ D4 ) @ ( times_times @ int @ C4 @ B5 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_divide_code
thf(fact_4413_rat__plus__code,axiom,
! [P: rat,Q2: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A7 @ D4 ) @ ( times_times @ int @ B5 @ C4 ) ) @ ( times_times @ int @ C4 @ D4 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_plus_code
thf(fact_4414_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_4415_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_4416_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_4417_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_4418_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_4419_these__insert__None,axiom,
! [A: $tType,A6: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( none @ A ) @ A6 ) )
= ( these @ A @ A6 ) ) ).
% these_insert_None
thf(fact_4420_take__Suc__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( take @ A @ ( suc @ N ) @ ( cons @ A @ X @ Xs2 ) )
= ( cons @ A @ X @ ( take @ A @ N @ Xs2 ) ) ) ).
% take_Suc_Cons
thf(fact_4421_take__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( nil @ A )
= ( take @ A @ N @ Xs2 ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs2
= ( nil @ A ) ) ) ) ).
% take_eq_Nil2
thf(fact_4422_take__eq__Nil,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( take @ A @ N @ Xs2 )
= ( nil @ A ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs2
= ( nil @ A ) ) ) ) ).
% take_eq_Nil
thf(fact_4423_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_4424_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_4425_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_4426_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_4427_these__insert__Some,axiom,
! [A: $tType,X: A,A6: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( some @ A @ X ) @ A6 ) )
= ( insert @ A @ X @ ( these @ A @ A6 ) ) ) ).
% these_insert_Some
thf(fact_4428_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_4429_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_4430_take__0,axiom,
! [A: $tType,Xs2: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs2 )
= ( nil @ A ) ) ).
% take_0
thf(fact_4431_in__these__eq,axiom,
! [A: $tType,X: A,A6: set @ ( option @ A )] :
( ( member @ A @ X @ ( these @ A @ A6 ) )
= ( member @ ( option @ A ) @ ( some @ A @ X ) @ A6 ) ) ).
% in_these_eq
thf(fact_4432_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_4433_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_4434_these__not__empty__eq,axiom,
! [A: $tType,B6: set @ ( option @ A )] :
( ( ( these @ A @ B6 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B6
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B6
!= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_4435_these__empty__eq,axiom,
! [A: $tType,B6: set @ ( option @ A )] :
( ( ( these @ A @ B6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B6
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B6
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_4436_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,C3: A,A3: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ 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_4437_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,A3: A,B2: A,C3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ 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_4438_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,A3: A,B2: A,C3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_4439_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,C3: A,A3: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C3 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_4440_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A8: set @ ( option @ A )] :
( image2 @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X4: option @ A] :
( ( member @ ( option @ A ) @ X4 @ A8 )
& ( X4
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_4441_take__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs2 ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs2 ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs2 ) ) ) ) ) ).
% take_Cons'
thf(fact_4442_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,N: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_minus'
thf(fact_4443_lex__take__index,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lex @ A @ R3 ) )
=> ~ ! [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 ) ) @ R3 ) ) ) ) ) ).
% lex_take_index
thf(fact_4444_take__Suc__conv__app__nth,axiom,
! [A: $tType,I2: nat,Xs2: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( take @ A @ ( suc @ I2 ) @ Xs2 )
= ( append @ A @ ( take @ A @ I2 @ Xs2 ) @ ( cons @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nil @ A ) ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_4445_nth__repl,axiom,
! [A: $tType,M2: nat,Xs2: list @ A,N: nat,X: 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 @ X @ ( nil @ A ) ) @ ( drop @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ Xs2 ) ) ) @ M2 )
= ( nth @ A @ Xs2 @ M2 ) ) ) ) ) ).
% nth_repl
thf(fact_4446_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_4447_upd__conv__take__nth__drop,axiom,
! [A: $tType,I2: nat,Xs2: list @ A,A3: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( list_update @ A @ Xs2 @ I2 @ A3 )
= ( append @ A @ ( take @ A @ I2 @ Xs2 ) @ ( cons @ A @ A3 @ ( drop @ A @ ( suc @ I2 ) @ Xs2 ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_4448_id__take__nth__drop,axiom,
! [A: $tType,I2: nat,Xs2: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( Xs2
= ( append @ A @ ( take @ A @ I2 @ Xs2 ) @ ( cons @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( drop @ A @ ( suc @ I2 ) @ Xs2 ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_4449_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X4: list @ A] : X4 ) ) ).
% drop0
thf(fact_4450_drop__Suc__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( drop @ A @ ( suc @ N ) @ ( cons @ A @ X @ Xs2 ) )
= ( drop @ A @ N @ Xs2 ) ) ).
% drop_Suc_Cons
thf(fact_4451_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_4452_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_4453_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_4454_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_4455_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_4456_nth__drop,axiom,
! [A: $tType,N: nat,Xs2: list @ A,I2: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs2 ) @ I2 )
= ( nth @ A @ Xs2 @ ( plus_plus @ nat @ N @ I2 ) ) ) ) ).
% nth_drop
thf(fact_4457_drop__0,axiom,
! [A: $tType,Xs2: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs2 )
= Xs2 ) ).
% drop_0
thf(fact_4458_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N5: nat,Xs: list @ A] : ( nths @ A @ Xs @ ( collect @ nat @ ( ord_less_eq @ nat @ N5 ) ) ) ) ) ).
% drop_eq_nths
thf(fact_4459_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_4460_drop__update__swap,axiom,
! [A: $tType,M2: nat,N: nat,Xs2: list @ A,X: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( drop @ A @ M2 @ ( list_update @ A @ Xs2 @ N @ X ) )
= ( list_update @ A @ ( drop @ A @ M2 @ Xs2 ) @ ( minus_minus @ nat @ N @ M2 ) @ X ) ) ) ).
% drop_update_swap
thf(fact_4461_append__eq__conv__conj,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs2 @ Ys )
= Zs )
= ( ( Xs2
= ( take @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ Zs ) )
& ( Ys
= ( drop @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_4462_drop__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs2 ) )
= ( cons @ A @ X @ Xs2 ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs2 ) )
= ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs2 ) ) ) ) ).
% drop_Cons'
thf(fact_4463_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_4464_Cons__nth__drop__Suc,axiom,
! [A: $tType,I2: nat,Xs2: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( cons @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( drop @ A @ ( suc @ I2 ) @ Xs2 ) )
= ( drop @ A @ I2 @ Xs2 ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_4465_zip__append2,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,Zs: list @ B] :
( ( zip @ A @ B @ Xs2 @ ( append @ B @ Ys @ Zs ) )
= ( 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 ) @ Zs ) ) ) ).
% zip_append2
thf(fact_4466_zip__append1,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ A,Zs: list @ B] :
( ( zip @ A @ B @ ( append @ A @ Xs2 @ Ys ) @ Zs )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ ( take @ B @ ( size_size @ ( list @ A ) @ Xs2 ) @ Zs ) ) @ ( zip @ A @ B @ Ys @ ( drop @ B @ ( size_size @ ( list @ A ) @ Xs2 ) @ Zs ) ) ) ) ).
% zip_append1
thf(fact_4467_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list @ A,I2: nat,J2: nat] :
( ( distinct @ A @ Vs )
=> ( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I2 @ Vs ) ) @ ( set2 @ A @ ( drop @ A @ J2 @ Vs ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_take_disj_set_drop_if_distinct
thf(fact_4468_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_4469_rotate__drop__take,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N5: nat,Xs: list @ A] : ( append @ A @ ( drop @ A @ ( modulo_modulo @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Xs ) @ ( take @ A @ ( modulo_modulo @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Xs ) ) ) ) ).
% rotate_drop_take
thf(fact_4470_dual__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( min @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 ) )
= ( ord_max @ A ) ) ) ).
% dual_min
thf(fact_4471_list__encode_Opelims,axiom,
! [X: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X )
=> ( ( ( X
= ( nil @ nat ) )
=> ( ( Y
= ( zero_zero @ nat ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
=> ~ ! [X3: nat,Xs3: list @ nat] :
( ( X
= ( cons @ nat @ X3 @ Xs3 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X3 @ Xs3 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_4472_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_4473_hd__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_4474_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_4475_hd__take,axiom,
! [A: $tType,J2: nat,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ J2 )
=> ( ( hd @ A @ ( take @ A @ J2 @ Xs2 ) )
= ( hd @ A @ Xs2 ) ) ) ).
% hd_take
thf(fact_4476_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_4477_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_4478_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_4479_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq: A > A > $o,A7: A,B5: A] : ( if @ A @ ( Less_eq @ A7 @ B5 ) @ A7 @ B5 ) ) ) ).
% ord.min_def
thf(fact_4480_ord_Omin_Ocong,axiom,
! [A: $tType] :
( ( min @ A )
= ( min @ A ) ) ).
% ord.min.cong
thf(fact_4481_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_4482_rotate__append,axiom,
! [A: $tType,L: list @ A,Q2: list @ A] :
( ( rotate @ A @ ( size_size @ ( list @ A ) @ L ) @ ( append @ A @ L @ Q2 ) )
= ( append @ A @ Q2 @ L ) ) ).
% rotate_append
thf(fact_4483_rotate__conv__mod,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N5: nat,Xs: list @ A] : ( rotate @ A @ ( modulo_modulo @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Xs ) ) ) ).
% rotate_conv_mod
thf(fact_4484_hd__conv__nth,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( hd @ A @ Xs2 )
= ( nth @ A @ Xs2 @ ( zero_zero @ nat ) ) ) ) ).
% hd_conv_nth
thf(fact_4485_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_4486_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_4487_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_4488_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_4489_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_4490_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_4491_tl__upt,axiom,
! [M2: nat,N: nat] :
( ( tl @ nat @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ N ) ) ).
% tl_upt
thf(fact_4492_length__upt,axiom,
! [I2: nat,J2: nat] :
( ( size_size @ ( list @ nat ) @ ( upt @ I2 @ J2 ) )
= ( minus_minus @ nat @ J2 @ I2 ) ) ).
% length_upt
thf(fact_4493_take__upt,axiom,
! [I2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ M2 ) @ N )
=> ( ( take @ nat @ M2 @ ( upt @ I2 @ N ) )
= ( upt @ I2 @ ( plus_plus @ nat @ I2 @ M2 ) ) ) ) ).
% take_upt
thf(fact_4494_upt__conv__Nil,axiom,
! [J2: nat,I2: nat] :
( ( ord_less_eq @ nat @ J2 @ I2 )
=> ( ( upt @ I2 @ J2 )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_4495_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_4496_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_4497_upt__eq__Nil__conv,axiom,
! [I2: nat,J2: nat] :
( ( ( upt @ I2 @ J2 )
= ( nil @ nat ) )
= ( ( J2
= ( zero_zero @ nat ) )
| ( ord_less_eq @ nat @ J2 @ I2 ) ) ) ).
% upt_eq_Nil_conv
thf(fact_4498_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N5: nat,M6: nat] : ( set2 @ nat @ ( upt @ N5 @ ( suc @ M6 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_4499_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_4500_Min__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( member @ A @ X @ A6 )
=> ( ( lattic643756798350308766er_Min @ A @ A6 )
= X ) ) ) ) ) ).
% Min_eqI
thf(fact_4501_Min__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X ) ) ) ) ).
% Min_le
thf(fact_4502_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N5: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N5 ) ) ) ) ).
% atLeast_upt
thf(fact_4503_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_4504_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_4505_upt__conv__Cons__Cons,axiom,
! [M2: nat,N: nat,Ns: list @ nat,Q2: nat] :
( ( ( cons @ nat @ M2 @ ( cons @ nat @ N @ Ns ) )
= ( upt @ M2 @ Q2 ) )
= ( ( cons @ nat @ N @ Ns )
= ( upt @ ( suc @ M2 ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_4506_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_4507_upt__0,axiom,
! [I2: nat] :
( ( upt @ I2 @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_4508_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N5: nat,M6: nat] : ( set2 @ nat @ ( upt @ ( suc @ N5 ) @ ( suc @ M6 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_4509_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N5: nat,M6: nat] : ( set2 @ nat @ ( upt @ ( suc @ N5 ) @ M6 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_4510_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N5: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N5 ) ) ) ) ) ).
% atMost_upto
thf(fact_4511_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( ord_less_eq @ A @ X @ A5 ) )
=> ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_4512_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
=> ! [A9: A] :
( ( member @ A @ A9 @ A6 )
=> ( ord_less_eq @ A @ X @ A9 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_4513_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_4514_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ord_less_eq @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_4515_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_4516_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 @ ( insert @ A @ A3 @ A6 ) )
= A3 ) ) ) ) ).
% Min_insert2
thf(fact_4517_upt__conv__Cons,axiom,
! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( upt @ I2 @ J2 )
= ( cons @ nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) ) ) ).
% upt_conv_Cons
thf(fact_4518_enumerate__eq__zip,axiom,
! [A: $tType] :
( ( enumerate @ A )
= ( ^ [N5: nat,Xs: list @ A] : ( zip @ nat @ A @ ( upt @ N5 @ ( plus_plus @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) ) ) @ Xs ) ) ) ).
% enumerate_eq_zip
thf(fact_4519_upt__add__eq__append,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( upt @ I2 @ ( plus_plus @ nat @ J2 @ K2 ) )
= ( append @ nat @ ( upt @ I2 @ J2 ) @ ( upt @ J2 @ ( plus_plus @ nat @ J2 @ K2 ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_4520_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B6 ) @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_4521_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M5: set @ A,N7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M5 @ N7 )
=> ( ( M5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N7 ) @ ( lattic643756798350308766er_Min @ A @ M5 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_4522_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_4523_remdups__adj__adjacent,axiom,
! [A: $tType,I2: nat,Xs2: list @ A] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs2 ) ) )
=> ( ( nth @ A @ ( remdups_adj @ A @ Xs2 ) @ I2 )
!= ( nth @ A @ ( remdups_adj @ A @ Xs2 ) @ ( suc @ I2 ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_4524_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_4525_remdups__adj__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_replicate
thf(fact_4526_remdups__adj__singleton,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ( ( remdups_adj @ A @ Xs2 )
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ( Xs2
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_4527_upt__rec,axiom,
( upt
= ( ^ [I: nat,J: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I @ J ) @ ( cons @ nat @ I @ ( upt @ ( suc @ I ) @ J ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_4528_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_4529_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_4530_upt__Suc,axiom,
! [I2: nat,J2: nat] :
( ( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( upt @ I2 @ ( suc @ J2 ) )
= ( append @ nat @ ( upt @ I2 @ J2 ) @ ( cons @ nat @ J2 @ ( nil @ nat ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( upt @ I2 @ ( suc @ J2 ) )
= ( nil @ nat ) ) ) ) ).
% upt_Suc
thf(fact_4531_upt__Suc__append,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( upt @ I2 @ ( suc @ J2 ) )
= ( append @ nat @ ( upt @ I2 @ J2 ) @ ( cons @ nat @ J2 @ ( nil @ nat ) ) ) ) ) ).
% upt_Suc_append
thf(fact_4532_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_4533_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 ) ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I )
= ( nth @ A @ Ys @ ( F4 @ I ) ) ) )
& ! [I: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ Xs2 @ I )
= ( nth @ A @ Xs2 @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) ) )
= ( ( F4 @ I )
= ( F4 @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_4534_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_4535_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_4536_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_4537_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_4538_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_4539_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_4540_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_mono @ nat @ A
@ ^ [I: nat] : ( compow @ ( A > A ) @ I @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_4541_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_4542_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,Y4: C] : ( product_Pair @ A @ B @ X4 @ ( F3 @ Y4 ) ) )
@ ( zip @ A @ C @ Xs2 @ Ys ) ) ) ).
% zip_map2
thf(fact_4543_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,Y4: D] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ Y4 ) ) )
@ ( zip @ C @ D @ Xs2 @ Ys ) ) ) ).
% zip_map_map
thf(fact_4544_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,Y4: C] : ( F3 @ ( product_Pair @ B @ C @ ( G3 @ X4 ) @ Y4 ) ) )
@ ( zip @ D @ C @ Xs2 @ Ys ) ) ) ).
% map_zip_map
thf(fact_4545_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,Y4: D] : ( F3 @ ( product_Pair @ B @ C @ X4 @ ( G3 @ Y4 ) ) ) )
@ ( zip @ B @ D @ Xs2 @ Ys ) ) ) ).
% map_zip_map2
thf(fact_4546_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ) ).
% mono_strict_invE
thf(fact_4547_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F4: A > B] :
! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) ) ) ) ) ).
% mono_def
thf(fact_4548_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( order_mono @ A @ B @ F3 ) ) ) ).
% monoI
thf(fact_4549_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% monoE
thf(fact_4550_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% monoD
thf(fact_4551_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_4552_mono__Suc,axiom,
order_mono @ nat @ nat @ suc ).
% mono_Suc
thf(fact_4553_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_4554_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% mono_add
thf(fact_4555_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_4556_incseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: nat > A,I2: nat] :
( ( order_mono @ nat @ A @ A6 )
=> ( ord_less_eq @ A @ ( A6 @ I2 ) @ ( A6 @ ( suc @ I2 ) ) ) ) ) ).
% incseq_SucD
thf(fact_4557_incseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
=> ( order_mono @ nat @ A @ X8 ) ) ) ).
% incseq_SucI
thf(fact_4558_incseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N5: nat] : ( ord_less_eq @ A @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_4559_incseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [X7: nat > A] :
! [M6: nat,N5: nat] :
( ( ord_less_eq @ nat @ M6 @ N5 )
=> ( ord_less_eq @ A @ ( X7 @ M6 ) @ ( X7 @ N5 ) ) ) ) ) ) ).
% incseq_def
thf(fact_4560_incseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,I2: nat,J2: nat] :
( ( order_mono @ nat @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( F3 @ J2 ) ) ) ) ) ).
% incseqD
thf(fact_4561_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mono_invE
thf(fact_4562_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F3: A > B,A6: A,B6: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( inf_inf @ A @ A6 @ B6 ) ) @ ( inf_inf @ B @ ( F3 @ A6 ) @ ( F3 @ B6 ) ) ) ) ) ).
% mono_inf
thf(fact_4563_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F3: A > B,A6: A,B6: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F3 @ A6 ) @ ( F3 @ B6 ) ) @ ( F3 @ ( sup_sup @ A @ A6 @ B6 ) ) ) ) ) ).
% mono_sup
thf(fact_4564_funpow__mono,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: A > A,A6: A,B6: A,N: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ A6 @ B6 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F3 @ A6 ) @ ( compow @ ( A > A ) @ N @ F3 @ B6 ) ) ) ) ) ).
% funpow_mono
thf(fact_4565_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_4566_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_4567_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_4568_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F3: A > B,M2: A,N: A,M4: B,N2: B] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ( image2 @ A @ B @ F3 @ ( set_or7035219750837199246ssThan @ A @ M2 @ N ) )
= ( set_or7035219750837199246ssThan @ B @ M4 @ N2 ) )
=> ( ( ord_less @ A @ M2 @ N )
=> ( ( F3 @ M2 )
= M4 ) ) ) ) ) ).
% mono_image_least
thf(fact_4569_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F3: A > A,P: A,K2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ P ) @ P )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K2 @ F3 @ ( bot_bot @ A ) ) @ P ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_4570_funpow__mono2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: A > A,I2: nat,J2: nat,X: A,Y: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ ( F3 @ X ) )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I2 @ F3 @ X ) @ ( compow @ ( A > A ) @ J2 @ F3 @ Y ) ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_4571_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,S3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ? [X5: A] :
( ( member @ A @ X5 @ S3 )
& ! [Xa4: A] :
( ( member @ A @ Xa4 @ S3 )
=> ( ord_less_eq @ A @ X5 @ Xa4 ) ) )
=> ( ( ord_Least @ B
@ ^ [Y4: B] : ( member @ B @ Y4 @ ( image2 @ A @ B @ F3 @ S3 ) ) )
= ( F3
@ ( ord_Least @ A
@ ^ [X4: A] : ( member @ A @ X4 @ S3 ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_4572_map__replicate__trivial,axiom,
! [A: $tType,X: A,I2: nat] :
( ( map @ nat @ A
@ ^ [I: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I2 ) )
= ( replicate @ A @ I2 @ X ) ) ).
% map_replicate_trivial
thf(fact_4573_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_4574_zip__eq__conv,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,Zs: list @ ( product_prod @ A @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( zip @ A @ B @ Xs2 @ Ys )
= Zs )
= ( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs )
= Xs2 )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_4575_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
@ ^ [I: nat] : ( F3 @ ( suc @ I ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_4576_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_4577_mono__ge2__power__minus__self,axiom,
! [K2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ( order_mono @ nat @ nat
@ ^ [M6: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K2 @ M6 ) @ M6 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_4578_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_4579_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_4580_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 )
@ ^ [I: nat] :
( map @ nat @ A
@ ^ [J: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs2 @ J ) @ I )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_4581_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs2: list @ A,X8: set @ A,F3: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ X8 )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs2 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( times_times @ nat @ ( count_list @ A @ Xs2 @ X4 ) @ ( F3 @ X4 ) )
@ X8 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_4582_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A] :
( ( finite_finite2 @ A @ ( image2 @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ F3 )
=> ( ! [N3: nat] :
( ( ( F3 @ N3 )
= ( F3 @ ( suc @ N3 ) ) )
=> ( ( F3 @ ( suc @ N3 ) )
= ( F3 @ ( suc @ ( suc @ N3 ) ) ) ) )
=> ? [N9: nat] :
( ! [N4: nat] :
( ( ord_less_eq @ nat @ N4 @ N9 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N9 )
=> ( ( ord_less @ nat @ M3 @ N4 )
=> ( ord_less @ A @ ( F3 @ M3 ) @ ( F3 @ N4 ) ) ) ) )
& ! [N4: nat] :
( ( ord_less_eq @ nat @ N9 @ N4 )
=> ( ( F3 @ N9 )
= ( F3 @ N4 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_4583_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_4584_atMost__UNIV__triv,axiom,
! [A: $tType] :
( ( set_ord_atMost @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% atMost_UNIV_triv
thf(fact_4585_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_4586_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_4587_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_4588_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_4589_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_4590_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_4591_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs2: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( zero_zero @ A )
@ Xs2 ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_4592_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_4593_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_4594_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_4595_not__UNIV__eq__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L3: A,H3: A] :
( ( top_top @ ( set @ A ) )
!= ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ).
% not_UNIV_eq_Icc
thf(fact_4596_atMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ( set_ord_atMost @ A @ X )
= ( top_top @ ( set @ A ) ) )
= ( X
= ( top_top @ A ) ) ) ) ).
% atMost_eq_UNIV_iff
thf(fact_4597_not__UNIV__eq__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H3: A] :
( ( top_top @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H3 ) ) ) ).
% not_UNIV_eq_Iic
thf(fact_4598_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_4599_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_4600_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_4601_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% top.extremum_strict
thf(fact_4602_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_4603_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( bounded_lattice @ A )
=> ! [X: A,Y: A] :
( ( ( set_or1337092689740270186AtMost @ A @ X @ Y )
= ( top_top @ ( set @ A ) ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_4604_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 )
@ ^ [Y4: A] : ( cons @ A @ Y4 @ Ys3 )
@ Xs2 )
@ ( n_lists @ A @ N @ Xs2 ) ) ) ) ).
% n_lists.simps(2)
thf(fact_4605_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_4606_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_4607_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs2 ) ) ) ) ).
% member_le_sum_list
thf(fact_4608_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_4609_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_4610_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_4611_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_4612_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( ( semiring_0 @ A )
=> ! [F3: B > A,C3: A,Xs2: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C3 )
@ Xs2 ) )
= ( times_times @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs2 ) ) @ C3 ) ) ) ).
% sum_list_mult_const
thf(fact_4613_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C3: A,F3: B > A,Xs2: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C3 @ ( F3 @ X4 ) )
@ Xs2 ) )
= ( times_times @ A @ C3 @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs2 ) ) ) ) ) ).
% sum_list_const_mult
thf(fact_4614_UNIV__option__conv,axiom,
! [A: $tType] :
( ( top_top @ ( set @ ( option @ A ) ) )
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% UNIV_option_conv
thf(fact_4615_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% sup_shunt
thf(fact_4616_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F3: A > A,P: A,K2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ P @ ( F3 @ P ) )
=> ( ord_less_eq @ A @ P @ ( compow @ ( A > A ) @ K2 @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_4617_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs2: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs2 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_4618_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs2: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ( ( 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_4619_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs2: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs2 ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_4620_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_4621_map__add__upt,axiom,
! [N: nat,M2: nat] :
( ( map @ nat @ nat
@ ^ [I: nat] : ( plus_plus @ nat @ I @ N )
@ ( upt @ ( zero_zero @ nat ) @ M2 ) )
= ( upt @ N @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% map_add_upt
thf(fact_4622_sum__list__replicate,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat,C3: A] :
( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N @ C3 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ C3 ) ) ) ).
% sum_list_replicate
thf(fact_4623_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs2 @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ ( product_prod @ B @ C ) ) )
@ ^ [X4: A,Y4: B,Z4: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C @ Y4 @ Z4 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs2 @ Ys ) @ Zs ) ) ) ).
% zip_assoc
thf(fact_4624_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs2 @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ B @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ B @ ( product_prod @ A @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [Y4: 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 @ Y4 @ Z4 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs2 @ Zs ) ) ) ) ).
% zip_left_commute
thf(fact_4625_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,Y4: A] : ( product_Pair @ A @ B @ Y4 @ X4 ) )
@ ( zip @ B @ A @ Ys3 @ Xs ) ) ) ) ).
% zip_commute
thf(fact_4626_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs2: list @ A,F3: A > B,G3: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( 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_4627_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_4628_notin__range__Some,axiom,
! [A: $tType,X: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_4629_zip__replicate1,axiom,
! [A: $tType,B: $tType,N: nat,X: A,Ys: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N @ X ) @ Ys )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ ( take @ B @ N @ Ys ) ) ) ).
% zip_replicate1
thf(fact_4630_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_4631_map__decr__upt,axiom,
! [M2: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( upt @ M2 @ N ) ) ).
% map_decr_upt
thf(fact_4632_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_4633_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_4634_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_4635_finite__UNIV__card__ge__0,axiom,
! [A: $tType] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_4636_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_4637_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs2: list @ A,Ys: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X @ Xs2 ) @ Ys )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys ) @ ( product @ A @ B @ Xs2 @ Ys ) ) ) ).
% product.simps(2)
thf(fact_4638_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R3: B,Xs2: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X4: C] : R3
@ Xs2 ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs2 ) ) @ R3 ) ) ) ).
% sum_list_triv
thf(fact_4639_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_4640_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F3: B > A] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_4641_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 )
@ ^ [Q5: nat] : ( product_Pair @ nat @ A @ Q5 @ A3 )
@ ( upt @ N @ ( plus_plus @ nat @ N @ M2 ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_4642_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_4643_card__length__sum__list__rec,axiom,
! [M2: nat,N7: 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 )
= N7 ) ) ) )
= ( 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 )
= N7 ) ) ) )
@ ( 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 ) )
= N7 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_4644_card__length__sum__list,axiom,
! [M2: nat,N7: nat] :
( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M2 )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N7 ) ) ) )
= ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N7 @ M2 ) @ ( one_one @ nat ) ) @ N7 ) ) ).
% card_length_sum_list
thf(fact_4645_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_4646_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image2 @ nat @ nat
@ ^ [M6: nat] : ( modulo_modulo @ nat @ M6 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_4647_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K2: nat,Xs2: list @ A,X: A] :
( ( ord_less @ nat @ K2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs2 @ K2 @ X ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs2 ) @ X ) @ ( nth @ A @ Xs2 @ K2 ) ) ) ) ) ).
% sum_list_update
thf(fact_4648_UNIV__nat__eq,axiom,
( ( top_top @ ( set @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UNIV_nat_eq
thf(fact_4649_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_4650_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F3: nat > B,Ns: list @ nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F3 @ Ns ) ) ) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
thf(fact_4651_range__mult,axiom,
! [A3: real] :
( ( ( A3
= ( zero_zero @ real ) )
=> ( ( image2 @ real @ real @ ( times_times @ real @ A3 ) @ ( top_top @ ( set @ real ) ) )
= ( insert @ real @ ( zero_zero @ real ) @ ( bot_bot @ ( set @ real ) ) ) ) )
& ( ( A3
!= ( zero_zero @ real ) )
=> ( ( image2 @ real @ real @ ( times_times @ real @ A3 ) @ ( top_top @ ( set @ real ) ) )
= ( top_top @ ( set @ real ) ) ) ) ) ).
% range_mult
thf(fact_4652_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P2: $o] :
( ( P2
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] : P2 ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P2
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] : P2 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_4653_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_4654_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_4655_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Zs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted2
thf(fact_4656_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_4657_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_4658_sorted__replicate,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: nat,X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N @ X ) ) ) ).
% sorted_replicate
thf(fact_4659_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_4660_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_4661_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_4662_sorted__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M2 @ N ) ) ).
% sorted_upt
thf(fact_4663_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_4664_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_4665_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,Y4: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
@ Xs2 ) ) ) ).
% sorted_map
thf(fact_4666_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_4667_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_4668_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_4669_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X @ X4 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) ) ) ) ).
% sorted_simps(2)
thf(fact_4670_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_4671_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_4672_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_4673_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 ) )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X4 @ Y4 ) ) ) ) ) ) ).
% sorted_append
thf(fact_4674_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_4675_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_4676_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_4677_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_4678_sorted__wrt__nth__less,axiom,
! [A: $tType,P2: A > A > $o,Xs2: list @ A,I2: nat,J2: nat] :
( ( sorted_wrt @ A @ P2 @ Xs2 )
=> ( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P2 @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J2 ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_4679_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P4: A > A > $o,Xs: list @ A] :
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P4 @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_4680_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_4681_sorted__wrt01,axiom,
! [A: $tType,Xs2: list @ A,P2: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P2 @ Xs2 ) ) ).
% sorted_wrt01
thf(fact_4682_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
= ( ! [I: nat,J: nat] :
( ( ord_less @ 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_iff_nth_mono_less
thf(fact_4683_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_4684_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ? [X3: list @ A] :
( ( ( set2 @ A @ X3 )
= A6 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X3 )
& ( distinct @ A @ X3 )
& ! [Y6: list @ A] :
( ( ( ( set2 @ A @ Y6 )
= A6 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y6 )
& ( distinct @ A @ Y6 ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_4685_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_4686_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
= ( ! [I: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I ) @ ( nth @ A @ Xs2 @ ( suc @ I ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_4687_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,I2: nat,J2: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J2 ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_4688_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
= ( ! [I: nat,J: nat] :
( ( 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_iff_nth_mono
thf(fact_4689_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_4690_sorted__wrt__less__idx,axiom,
! [Ns: list @ nat,I2: nat] :
( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ nat ) @ Ns ) )
=> ( ord_less_eq @ nat @ I2 @ ( nth @ nat @ Ns @ I2 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_4691_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,P2: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P2 @ X5 ) )
=> ( ( find @ A @ P2 @ Xs2 )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
& ( P2 @ X4 ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_4692_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_4693_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_4694_root__def,axiom,
( root
= ( ^ [N5: nat,X4: real] :
( if @ real
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y4: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y4 ) @ ( power_power @ real @ ( abs_abs @ real @ Y4 ) @ N5 ) )
@ X4 ) ) ) ) ).
% root_def
thf(fact_4695_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_4696_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_4697_length__rev,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_rev
thf(fact_4698_top2I,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : ( top_top @ ( A > B > $o ) @ X @ Y ) ).
% top2I
thf(fact_4699_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_4700_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_4701_rev__drop,axiom,
! [A: $tType,I2: nat,Xs2: list @ A] :
( ( rev @ A @ ( drop @ A @ I2 @ Xs2 ) )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ I2 ) @ ( rev @ A @ Xs2 ) ) ) ).
% rev_drop
thf(fact_4702_rev__take,axiom,
! [A: $tType,I2: nat,Xs2: list @ A] :
( ( rev @ A @ ( take @ A @ I2 @ Xs2 ) )
= ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ I2 ) @ ( rev @ A @ Xs2 ) ) ) ).
% rev_take
thf(fact_4703_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_4704_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_4705_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_4706_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_4707_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_4708_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_4709_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs2 ) )
= ( ! [I: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ ( suc @ I ) ) @ ( nth @ A @ Xs2 @ I ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_4710_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs2 ) )
= ( ! [I: nat,J: nat] :
( ( 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_iff_nth_mono
thf(fact_4711_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,I2: nat,J2: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J2 ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_4712_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),I2: nat,J2: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) ) )
=> ( ( ord_less @ nat @ J2
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs2 ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs2 ) @ I2 ) @ J2 )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs2 @ J2 ) @ I2 ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_4713_transpose__column,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),I2: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I2 )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs2 ) ) )
= ( nth @ ( list @ A ) @ Xs2 @ I2 ) ) ) ) ).
% transpose_column
thf(fact_4714_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_4715_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_4716_length__filter__le,axiom,
! [A: $tType,P2: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_filter_le
thf(fact_4717_sum__length__filter__compl,axiom,
! [A: $tType,P2: A > $o,Xs2: list @ A] :
( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ Xs2 ) )
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ^ [X4: A] :
~ ( P2 @ X4 )
@ Xs2 ) ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% sum_length_filter_compl
thf(fact_4718_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_4719_replicate__length__filter,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X )
@ Xs2 ) )
@ X )
= ( filter2 @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X )
@ Xs2 ) ) ).
% replicate_length_filter
thf(fact_4720_length__filter__less,axiom,
! [A: $tType,X: A,Xs2: list @ A,P2: A > $o] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ~ ( P2 @ X )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% length_filter_less
thf(fact_4721_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B,P2: 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 @ P2 @ Xs2 ) ) ) ) ) ).
% sorted_filter
thf(fact_4722_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_4723_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F3: B > A,P2: B > $o,Xs2: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs2 ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P2 @ X4 ) @ ( F3 @ X4 ) @ ( zero_zero @ A ) )
@ Xs2 ) ) ) ) ).
% sum_list_map_filter'
thf(fact_4724_sum__list__filter__le__nat,axiom,
! [A: $tType,F3: A > nat,P2: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ ( filter2 @ A @ P2 @ Xs2 ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs2 ) ) ) ).
% sum_list_filter_le_nat
thf(fact_4725_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs2: list @ B,P2: B > $o,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs2 ) )
=> ( ~ ( P2 @ X3 )
=> ( ( F3 @ X3 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs2 ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs2 ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_4726_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P4: A > $o,Xs: list @ A] :
( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P4 @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_4727_length__filter__conv__card,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P @ ( nth @ A @ Xs2 @ I ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_4728_distinct__length__filter,axiom,
! [A: $tType,Xs2: list @ A,P2: A > $o] :
( ( distinct @ A @ Xs2 )
=> ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ Xs2 ) )
= ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% distinct_length_filter
thf(fact_4729_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_4730_nth__transpose,axiom,
! [A: $tType,I2: nat,Xs2: list @ ( list @ A )] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs2 ) @ I2 )
= ( map @ ( list @ A ) @ A
@ ^ [Xs: list @ A] : ( nth @ A @ Xs @ I2 )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs2 ) ) ) ) ).
% nth_transpose
thf(fact_4731_transpose__column__length,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),I2: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs2 ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ I2 ) ) ) ) ) ).
% transpose_column_length
thf(fact_4732_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_4733_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F3: B > A,P2: B > $o,Xs2: list @ B] :
( ( map @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs2 ) )
= ( map_filter @ B @ A
@ ^ [X4: B] : ( if @ ( option @ A ) @ ( P2 @ X4 ) @ ( some @ A @ ( F3 @ X4 ) ) @ ( none @ A ) )
@ Xs2 ) ) ).
% map_filter_map_filter
thf(fact_4734_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_4735_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_4736_nths__shift__lemma__Suc,axiom,
! [A: $tType,P2: 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] : ( P2 @ ( 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] : ( P2 @ ( product_snd @ A @ nat @ P5 ) )
@ ( zip @ A @ nat @ Xs2 @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_4737_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A7: A,Xs: list @ B] :
( foldr @ B @ A
@ ^ [X4: B,B5: A] : ( plus_plus @ A @ ( F4 @ X4 ) @ ( times_times @ A @ A7 @ B5 ) )
@ Xs
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_4738_nths__shift__lemma,axiom,
! [A: $tType,A6: set @ nat,Xs2: list @ A,I2: 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 @ I2 @ ( plus_plus @ nat @ I2 @ ( 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 ) @ I2 ) @ A6 )
@ ( zip @ A @ nat @ Xs2 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_4739_nths__def,axiom,
! [A: $tType] :
( ( nths @ A )
= ( ^ [Xs: list @ A,A8: 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 ) @ A8 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ) ).
% nths_def
thf(fact_4740_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_4741_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_4742_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_4743_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_4744_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% map_of_eq_Some_iff
thf(fact_4745_map__of__zip__is__None,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) @ X )
= ( none @ B ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_4746_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) )
=> ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y ) ) ) ) ).
% map_of_is_SomeI
thf(fact_4747_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y: B,X: A] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( some @ B @ Y )
= ( map_of @ A @ B @ Xys @ X ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% Some_eq_map_of_iff
thf(fact_4748_length__takeWhile__le,axiom,
! [A: $tType,P2: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_takeWhile_le
thf(fact_4749_sorted__takeWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,P2: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P2 @ Xs2 ) ) ) ) ).
% sorted_takeWhile
thf(fact_4750_map__of__filter__in,axiom,
! [B: $tType,A: $tType,Xs2: list @ ( product_prod @ B @ A ),K2: B,Z3: A,P2: B > A > $o] :
( ( ( map_of @ B @ A @ Xs2 @ K2 )
= ( some @ A @ Z3 ) )
=> ( ( P2 @ K2 @ Z3 )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P2 ) @ Xs2 ) @ K2 )
= ( some @ A @ Z3 ) ) ) ) ).
% map_of_filter_in
thf(fact_4751_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_4752_takeWhile__nth,axiom,
! [A: $tType,J2: nat,P2: A > $o,Xs2: list @ A] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs2 ) ) )
=> ( ( nth @ A @ ( takeWhile @ A @ P2 @ Xs2 ) @ J2 )
= ( nth @ A @ Xs2 @ J2 ) ) ) ).
% takeWhile_nth
thf(fact_4753_nth__length__takeWhile,axiom,
! [A: $tType,P2: A > $o,Xs2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ~ ( P2 @ ( nth @ A @ Xs2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs2 ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_4754_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_4755_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K2: A,X: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ X ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X3: B] :
( ( map_of @ A @ B @ L @ K2 )
= ( some @ B @ X3 ) ) ) ).
% weak_map_of_SomeI
thf(fact_4756_map__of__zip__inject,axiom,
! [B: $tType,A: $tType,Ys: list @ A,Xs2: list @ B,Zs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys )
= ( size_size @ ( list @ B ) @ Xs2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( 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 @ Zs ) ) )
=> ( Ys = Zs ) ) ) ) ) ).
% map_of_zip_inject
thf(fact_4757_map__of__Cons__code_I2_J,axiom,
! [C: $tType,B: $tType,L: B,K2: B,V3: C,Ps: list @ ( product_prod @ B @ C )] :
( ( ( L = K2 )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V3 ) @ Ps ) @ K2 )
= ( some @ C @ V3 ) ) )
& ( ( L != K2 )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V3 ) @ Ps ) @ K2 )
= ( map_of @ B @ C @ Ps @ K2 ) ) ) ) ).
% map_of_Cons_code(2)
thf(fact_4758_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J2: nat,P2: A > $o,Xs2: list @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( P2 @ ( nth @ A @ Xs2 @ I3 ) ) )
=> ( ( ord_less_eq @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ nat @ J2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs2 ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_4759_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs2: list @ A,P2: A > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P2 @ ( nth @ A @ Xs2 @ I3 ) ) ) )
=> ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ~ ( P2 @ ( nth @ A @ Xs2 @ N ) ) )
=> ( ( takeWhile @ A @ P2 @ Xs2 )
= ( take @ A @ N @ Xs2 ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_4760_map__of__zip__is__Some,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
= ( ? [Y4: B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) @ X )
= ( some @ B @ Y4 ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_4761_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_4762_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,I2: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs2 )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) @ ( nth @ A @ Xs2 @ I2 ) )
= ( some @ B @ ( nth @ B @ Ys @ I2 ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_4763_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_4764_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,V4: B] :
( ( map_of @ A @ B @ Xs2 @ K3 )
= ( some @ B @ V4 ) ) ) ) ) ) ).
% set_map_of_compr
thf(fact_4765_DERIV__even__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_4766_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D5: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D5
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D5
= ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( D5
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ D5 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_4767_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X9: real] :
( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X9 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) )
@ ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( power_power @ real @ X @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_arctan_series
thf(fact_4768_has__field__derivative__sinh,axiom,
! [A10: $tType] :
( ( ( real_Vector_banach @ A10 )
& ( real_V3459762299906320749_field @ A10 ) )
=> ! [G3: A10 > A10,Db: A10,X: A10,S2: set @ A10] :
( ( has_field_derivative @ A10 @ G3 @ Db @ ( topolo174197925503356063within @ A10 @ X @ S2 ) )
=> ( has_field_derivative @ A10
@ ^ [X4: A10] : ( sinh @ A10 @ ( G3 @ X4 ) )
@ ( times_times @ A10 @ ( cosh @ A10 @ ( G3 @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A10 @ X @ S2 ) ) ) ) ).
% has_field_derivative_sinh
thf(fact_4769_has__field__derivative__cosh,axiom,
! [A10: $tType] :
( ( ( real_Vector_banach @ A10 )
& ( real_V3459762299906320749_field @ A10 ) )
=> ! [G3: A10 > A10,Db: A10,X: A10,S2: set @ A10] :
( ( has_field_derivative @ A10 @ G3 @ Db @ ( topolo174197925503356063within @ A10 @ X @ S2 ) )
=> ( has_field_derivative @ A10
@ ^ [X4: A10] : ( cosh @ A10 @ ( G3 @ X4 ) )
@ ( times_times @ A10 @ ( sinh @ A10 @ ( G3 @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A10 @ X @ S2 ) ) ) ) ).
% has_field_derivative_cosh
thf(fact_4770_DERIV__fun__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G3: A > A,M2: A,X: A] :
( ( has_field_derivative @ A @ G3 @ M2 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( exp @ A @ ( G3 @ X4 ) )
@ ( times_times @ A @ ( exp @ A @ ( G3 @ X ) ) @ M2 )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_exp
thf(fact_4771_DERIV__fun__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G3: A > A,M2: A,X: A] :
( ( has_field_derivative @ A @ G3 @ M2 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( sin @ A @ ( G3 @ X4 ) )
@ ( times_times @ A @ ( cos @ A @ ( G3 @ X ) ) @ M2 )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_sin
thf(fact_4772_DERIV__fun__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G3: A > A,M2: A,X: A] :
( ( has_field_derivative @ A @ G3 @ M2 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( cos @ A @ ( G3 @ X4 ) )
@ ( times_times @ A @ ( uminus_uminus @ A @ ( sin @ A @ ( G3 @ X ) ) ) @ M2 )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_cos
thf(fact_4773_termdiffs__strong__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,X: A] :
( ! [Y3: A] :
( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ Y3 @ N5 ) ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ X4 @ N5 ) ) )
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% termdiffs_strong_converges_everywhere
thf(fact_4774_DERIV__fun__pow,axiom,
! [G3: real > real,M2: real,X: real,N: nat] :
( ( has_field_derivative @ real @ G3 @ M2 @ ( topolo174197925503356063within @ real @ X @ ( 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 @ X ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) @ M2 )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_fun_pow
thf(fact_4775_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K5: real,C3: nat > A,F3: A > A,F6: A,Z3: A] :
( ! [Z: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K5 )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ Z @ N5 ) )
@ ( F3 @ Z ) ) )
=> ( ( has_field_derivative @ A @ F3 @ F6 @ ( topolo174197925503356063within @ A @ Z3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N5 ) @ ( power_power @ A @ Z3 @ N5 ) )
@ F6 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_4776_has__real__derivative__powr,axiom,
! [Z3: real,R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z3 )
=> ( has_field_derivative @ real
@ ^ [Z4: real] : ( powr @ real @ Z4 @ R3 )
@ ( times_times @ real @ R3 @ ( powr @ real @ Z3 @ ( minus_minus @ real @ R3 @ ( one_one @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% has_real_derivative_powr
thf(fact_4777_DERIV__series_H,axiom,
! [F3: real > nat > real,F6: real > nat > real,X0: real,A3: real,B2: real,L5: nat > real] :
( ! [N3: nat] :
( has_field_derivative @ real
@ ^ [X4: real] : ( F3 @ X4 @ N3 )
@ ( F6 @ X0 @ N3 )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B2 ) )
=> ( summable @ real @ ( F3 @ X3 ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B2 ) )
=> ( ( summable @ real @ ( F6 @ X0 ) )
=> ( ( summable @ real @ L5 )
=> ( ! [N3: nat,X3: real,Y3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B2 ) )
=> ( ( member @ real @ Y3 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B2 ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( F3 @ X3 @ N3 ) @ ( F3 @ Y3 @ N3 ) ) ) @ ( times_times @ real @ ( L5 @ N3 ) @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y3 ) ) ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X4: real] : ( suminf @ real @ ( F3 @ X4 ) )
@ ( suminf @ real @ ( F6 @ X0 ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_4778_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K5: real,C3: nat > A,Z3: A] :
( ! [Z: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K5 )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ Z @ N5 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( has_field_derivative @ A
@ ^ [Z4: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ Z4 @ N5 ) ) )
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N5 ) @ ( power_power @ A @ Z3 @ N5 ) ) )
@ ( topolo174197925503356063within @ A @ Z3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_4779_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K5: A,X: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ K5 @ N5 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ X4 @ N5 ) ) )
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_4780_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K5: A,X: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ K5 @ N5 ) ) )
=> ( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N5 ) @ ( power_power @ A @ K5 @ N5 ) ) )
=> ( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C3 ) @ N5 ) @ ( power_power @ A @ K5 @ N5 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ X4 @ N5 ) ) )
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_4781_DERIV__log,axiom,
! [X: real,B2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( log2 @ B2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B2 ) @ X ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_log
thf(fact_4782_DERIV__fun__powr,axiom,
! [G3: real > real,M2: real,X: real,R3: real] :
( ( has_field_derivative @ real @ G3 @ M2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( has_field_derivative @ real
@ ^ [X4: real] : ( powr @ real @ ( G3 @ X4 ) @ R3 )
@ ( times_times @ real @ ( times_times @ real @ R3 @ ( powr @ real @ ( G3 @ X ) @ ( minus_minus @ real @ R3 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M2 )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_fun_powr
thf(fact_4783_DERIV__powr,axiom,
! [G3: real > real,M2: real,X: real,F3: real > real,R3: real] :
( ( has_field_derivative @ real @ G3 @ M2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( ( has_field_derivative @ real @ F3 @ R3 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X4: real] : ( powr @ real @ ( G3 @ X4 ) @ ( F3 @ X4 ) )
@ ( times_times @ real @ ( powr @ real @ ( G3 @ X ) @ ( F3 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ R3 @ ( ln_ln @ real @ ( G3 @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M2 @ ( F3 @ X ) ) @ ( G3 @ X ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_powr
thf(fact_4784_DERIV__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( tan @ A ) @ ( inverse_inverse @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_tan
thf(fact_4785_DERIV__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( cot @ A ) @ ( uminus_uminus @ A @ ( inverse_inverse @ A @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_cot
thf(fact_4786_has__field__derivative__tanh,axiom,
! [A10: $tType] :
( ( ( real_Vector_banach @ A10 )
& ( real_V3459762299906320749_field @ A10 ) )
=> ! [G3: A10 > A10,X: A10,Db: A10,S2: set @ A10] :
( ( ( cosh @ A10 @ ( G3 @ X ) )
!= ( zero_zero @ A10 ) )
=> ( ( has_field_derivative @ A10 @ G3 @ Db @ ( topolo174197925503356063within @ A10 @ X @ S2 ) )
=> ( has_field_derivative @ A10
@ ^ [X4: A10] : ( tanh @ A10 @ ( G3 @ X4 ) )
@ ( times_times @ A10 @ ( minus_minus @ A10 @ ( one_one @ A10 ) @ ( power_power @ A10 @ ( tanh @ A10 @ ( G3 @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A10 @ X @ S2 ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_4787_DERIV__power__series_H,axiom,
! [R2: real,F3: nat > real,X0: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R2 ) @ R2 ) )
=> ( summable @ real
@ ^ [N5: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N5 ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) @ ( power_power @ real @ X3 @ N5 ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R2 ) @ R2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( has_field_derivative @ real
@ ^ [X4: real] :
( suminf @ real
@ ^ [N5: nat] : ( times_times @ real @ ( F3 @ N5 ) @ ( power_power @ real @ X4 @ ( suc @ N5 ) ) ) )
@ ( suminf @ real
@ ^ [N5: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N5 ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) @ ( power_power @ real @ X0 @ N5 ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_4788_DERIV__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_real_root
thf(fact_4789_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F3: real > real,X: real,N: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ X @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_4790_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F3: real > real,X: real,N: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
& ! [M: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ X @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_4791_DERIV__odd__real__root,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_4792_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,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ H @ T7 )
& ( ord_less_eq @ real @ T7 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ H @ T7 )
& ( ord_less @ real @ T7 @ ( zero_zero @ real ) )
& ( ( F3 @ H )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ H @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_4793_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,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H )
& ( ( F3 @ H )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ H @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_4794_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,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ H )
& ( ( F3 @ H )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ H @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_4795_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ! [M: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ X @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_4796_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ X @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_4797_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,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ A3 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A3 @ C3 )
=> ( ( ord_less_eq @ real @ C3 @ B2 )
=> ? [T7: real] :
( ( ord_less @ real @ A3 @ T7 )
& ( ord_less @ real @ T7 @ C3 )
& ( ( F3 @ A3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ C3 ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A3 @ C3 ) @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ A3 @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_4798_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,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ A3 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ C3 )
=> ( ( ord_less @ real @ C3 @ B2 )
=> ? [T7: real] :
( ( ord_less @ real @ C3 @ T7 )
& ( ord_less @ real @ T7 @ B2 )
& ( ( F3 @ B2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ C3 ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C3 ) @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_4799_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A3: real,B2: real,C3: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ A3 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ C3 )
=> ( ( ord_less_eq @ real @ C3 @ B2 )
=> ( ( ord_less_eq @ real @ A3 @ X )
=> ( ( ord_less_eq @ real @ X @ B2 )
=> ( ( X != C3 )
=> ? [T7: real] :
( ( ( ord_less @ real @ X @ C3 )
=> ( ( ord_less @ real @ X @ T7 )
& ( ord_less @ real @ T7 @ C3 ) ) )
& ( ~ ( ord_less @ real @ X @ C3 )
=> ( ( ord_less @ real @ C3 @ T7 )
& ( ord_less @ real @ T7 @ X ) ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M6 @ C3 ) @ ( semiring_char_0_fact @ real @ M6 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C3 ) @ M6 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_4800_Maclaurin__lemma2,axiom,
! [N: nat,H: real,Diff: nat > real > real,K2: nat,B6: real] :
( ! [M: nat,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N
= ( suc @ K2 ) )
=> ! [M3: nat,T8: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H ) )
=> ( has_field_derivative @ real
@ ^ [U2: real] :
( minus_minus @ real @ ( Diff @ M3 @ U2 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M3 @ P5 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ real @ U2 @ P5 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ M3 ) ) )
@ ( times_times @ real @ B6 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N @ M3 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ M3 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M3 ) @ T8 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M3 ) @ P5 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P5 ) ) @ ( power_power @ real @ T8 @ P5 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) )
@ ( times_times @ real @ B6 @ ( divide_divide @ real @ ( power_power @ real @ T8 @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_4801_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( F3 @ X )
!= ( 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 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_4802_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A,G3: A > A,E3: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_field_derivative @ A @ G3 @ E3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( G3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y4: A] : ( divide_divide @ A @ ( F3 @ Y4 ) @ ( G3 @ Y4 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D3 @ ( G3 @ X ) ) @ ( times_times @ A @ E3 @ ( F3 @ X ) ) ) @ ( power_power @ A @ ( G3 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_4803_DERIV__pow,axiom,
! [N: nat,X: real,S2: 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 @ X @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ S2 ) ) ).
% DERIV_pow
thf(fact_4804_mult__commute__abs,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [C3: A] :
( ( ^ [X4: A] : ( times_times @ A @ X4 @ C3 ) )
= ( times_times @ A @ C3 ) ) ) ).
% mult_commute_abs
thf(fact_4805_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K2: A,F5: filter @ A] :
( has_field_derivative @ A
@ ^ [X4: A] : K2
@ ( zero_zero @ A )
@ F5 ) ) ).
% DERIV_const
thf(fact_4806_DERIV__cmult__Id,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,X: A,S2: set @ A] : ( has_field_derivative @ A @ ( times_times @ A @ C3 ) @ C3 @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ).
% DERIV_cmult_Id
thf(fact_4807_DERIV__cmult__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A,S2: set @ A,C3: A] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( times_times @ A @ ( F3 @ X4 ) @ C3 )
@ ( times_times @ A @ D5 @ C3 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_cmult_right
thf(fact_4808_DERIV__cmult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A,S2: set @ A,C3: A] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( times_times @ A @ C3 @ ( F3 @ X4 ) )
@ ( times_times @ A @ C3 @ D5 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_cmult
thf(fact_4809_DERIV__mult_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A,S2: set @ A,G3: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_field_derivative @ A @ G3 @ E5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F3 @ X ) @ E5 ) @ ( times_times @ A @ D5 @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_mult'
thf(fact_4810_DERIV__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Da: A,X: A,S2: set @ A,G3: A > A,Db: A] :
( ( has_field_derivative @ A @ F3 @ Da @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_field_derivative @ A @ G3 @ Db @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G3 @ X ) ) @ ( times_times @ A @ Db @ ( F3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_mult
thf(fact_4811_DERIV__chain_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A,S2: set @ A,G3: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_field_derivative @ A @ G3 @ E5 @ ( topolo174197925503356063within @ A @ ( F3 @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( G3 @ ( F3 @ X4 ) )
@ ( times_times @ A @ E5 @ D5 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_chain'
thf(fact_4812_DERIV__chain2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Da: A,G3: A > A,X: A,Db: A,S2: set @ A] :
( ( has_field_derivative @ A @ F3 @ Da @ ( topolo174197925503356063within @ A @ ( G3 @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ G3 @ Db @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( F3 @ ( G3 @ X4 ) )
@ ( times_times @ A @ Da @ Db )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_chain2
thf(fact_4813_DERIV__chain3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [G3: A > A,G6: A > A,F3: A > A,F6: A,X: A] :
( ! [X3: A] : ( has_field_derivative @ A @ G3 @ ( G6 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ F3 @ F6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( G3 @ ( F3 @ X4 ) )
@ ( times_times @ A @ F6 @ ( G6 @ ( F3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% DERIV_chain3
thf(fact_4814_DERIV__chain__s,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [S2: set @ A,G3: A > A,G6: A > A,F3: A > A,F6: A,X: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( has_field_derivative @ A @ G3 @ ( G6 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) )
=> ( ( has_field_derivative @ A @ F3 @ F6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ A @ ( F3 @ X ) @ S2 )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( G3 @ ( F3 @ X4 ) )
@ ( times_times @ A @ F6 @ ( G6 @ ( F3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% DERIV_chain_s
thf(fact_4815_DERIV__const__ratio__const,axiom,
! [A3: real,B2: real,F3: real > real,K2: real] :
( ( A3 != B2 )
=> ( ! [X3: real] : ( has_field_derivative @ real @ F3 @ K2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A3 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A3 ) @ K2 ) ) ) ) ).
% DERIV_const_ratio_const
thf(fact_4816_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A,S2: set @ A,G3: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( has_field_derivative @ A @ G3 @ E5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( G3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D5 @ ( G3 @ X ) ) @ ( times_times @ A @ ( F3 @ X ) @ E5 ) ) @ ( times_times @ A @ ( G3 @ X ) @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_4817_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A,S2: set @ A] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F3 @ X ) ) @ D5 ) @ ( inverse_inverse @ A @ ( F3 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_4818_MVT2,axiom,
! [A3: real,B2: real,F3: real > real,F6: real > real] :
( ( ord_less @ real @ A3 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ( has_field_derivative @ real @ F3 @ ( F6 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z: real] :
( ( ord_less @ real @ A3 @ Z )
& ( ord_less @ real @ Z @ B2 )
& ( ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A3 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A3 ) @ ( F6 @ Z ) ) ) ) ) ) ).
% MVT2
thf(fact_4819_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A,S2: set @ A,N: nat] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( 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 @ D5 @ ( power_power @ A @ ( F3 @ X ) @ N ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_power_Suc
thf(fact_4820_DERIV__inverse,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,S2: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_inverse
thf(fact_4821_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A,S2: set @ A,N: nat] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( power_power @ A @ ( F3 @ X4 ) @ N )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( times_times @ A @ D5 @ ( power_power @ A @ ( F3 @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_power
thf(fact_4822_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X: A,G6: A > real,S2: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G3 @ X ) )
=> ( ( ord_less @ real @ ( G3 @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( 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 @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_4823_has__derivative__tan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X: A,G6: A > real,S2: set @ A] :
( ( ( cos @ real @ ( G3 @ X ) )
!= ( zero_zero @ real ) )
=> ( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( 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 @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivative_tan
thf(fact_4824_has__derivative__arctan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,G6: A > real,X: A,S2: set @ A] :
( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( 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 @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_arctan
thf(fact_4825_has__derivative__mult__right,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G3: C > A,G6: C > A,F5: filter @ C,X: A] :
( ( has_derivative @ C @ A @ G3 @ G6 @ F5 )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( times_times @ A @ X @ ( G3 @ X4 ) )
@ ^ [X4: C] : ( times_times @ A @ X @ ( G6 @ X4 ) )
@ F5 ) ) ) ).
% has_derivative_mult_right
thf(fact_4826_has__derivative__mult__left,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G3: C > A,G6: C > A,F5: filter @ C,Y: A] :
( ( has_derivative @ C @ A @ G3 @ G6 @ F5 )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( times_times @ A @ ( G3 @ X4 ) @ Y )
@ ^ [X4: C] : ( times_times @ A @ ( G6 @ X4 ) @ Y )
@ F5 ) ) ) ).
% has_derivative_mult_left
thf(fact_4827_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C3: B,F5: filter @ A] :
( has_derivative @ A @ B
@ ^ [X4: A] : C3
@ ^ [X4: A] : ( zero_zero @ B )
@ F5 ) ) ).
% has_derivative_const
thf(fact_4828_has__field__derivative__imp__has__derivative,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,F5: filter @ A] :
( ( has_field_derivative @ A @ F3 @ D5 @ F5 )
=> ( has_derivative @ A @ A @ F3 @ ( times_times @ A @ D5 ) @ F5 ) ) ) ).
% has_field_derivative_imp_has_derivative
thf(fact_4829_has__derivative__imp__has__field__derivative,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A > A,F5: filter @ A,D7: A] :
( ( has_derivative @ A @ A @ F3 @ D5 @ F5 )
=> ( ! [X3: A] :
( ( times_times @ A @ X3 @ D7 )
= ( D5 @ X3 ) )
=> ( has_field_derivative @ A @ F3 @ D7 @ F5 ) ) ) ) ).
% has_derivative_imp_has_field_derivative
thf(fact_4830_has__field__derivative__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( ( has_field_derivative @ A )
= ( ^ [F4: A > A,D8: A] : ( has_derivative @ A @ A @ F4 @ ( times_times @ A @ D8 ) ) ) ) ) ).
% has_field_derivative_def
thf(fact_4831_has__derivative__mult,axiom,
! [A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F3: D > A,F6: D > A,X: D,S2: set @ D,G3: D > A,G6: D > A] :
( ( has_derivative @ D @ A @ F3 @ F6 @ ( topolo174197925503356063within @ D @ X @ S2 ) )
=> ( ( has_derivative @ D @ A @ G3 @ G6 @ ( topolo174197925503356063within @ D @ X @ S2 ) )
=> ( has_derivative @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ^ [H2: D] : ( plus_plus @ A @ ( times_times @ A @ ( F3 @ X ) @ ( G6 @ H2 ) ) @ ( times_times @ A @ ( F6 @ H2 ) @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ D @ X @ S2 ) ) ) ) ) ).
% has_derivative_mult
thf(fact_4832_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F5: A > B,X: A] :
( ( has_derivative @ A @ B
@ ^ [X4: A] : ( zero_zero @ B )
@ F5
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( F5
= ( ^ [H2: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_4833_has__derivative__exp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,G6: A > real,X: A,S2: set @ A] :
( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( exp @ real @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G6 @ X4 ) @ ( exp @ real @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_exp
thf(fact_4834_has__derivative__sin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,G6: A > real,X: A,S2: set @ A] :
( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( sin @ real @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G6 @ X4 ) @ ( cos @ real @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_sin
thf(fact_4835_has__derivative__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G3: A > A,Db: A,X: A,S2: set @ A] :
( ( has_derivative @ A @ A @ G3 @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ A
@ ^ [X4: A] : ( sinh @ A @ ( G3 @ X4 ) )
@ ( times_times @ A @ ( times_times @ A @ ( cosh @ A @ ( G3 @ X ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_sinh
thf(fact_4836_has__derivative__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G3: A > A,Db: A,X: A,S2: set @ A] :
( ( has_derivative @ A @ A @ G3 @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ A
@ ^ [X4: A] : ( cosh @ A @ ( G3 @ X4 ) )
@ ( times_times @ A @ ( times_times @ A @ ( sinh @ A @ ( G3 @ X ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_cosh
thf(fact_4837_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,F6: C > A,X: C,S3: set @ C,G3: C > A,G6: C > A] :
( ( has_derivative @ C @ A @ F3 @ F6 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( has_derivative @ C @ A @ G3 @ G6 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( ( G3 @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ^ [H2: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F6 @ H2 ) @ ( G3 @ X ) ) @ ( times_times @ A @ ( F3 @ X ) @ ( G6 @ H2 ) ) ) @ ( times_times @ A @ ( G3 @ X ) @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_4838_has__derivative__inverse_H,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,S3: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A @ ( inverse_inverse @ A )
@ ^ [H2: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X ) @ H2 ) @ ( inverse_inverse @ A @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_inverse'
thf(fact_4839_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: C > A,X: C,F6: C > A,S3: set @ C] :
( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F3 @ F6 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ ^ [H2: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F3 @ X ) ) @ ( F6 @ H2 ) ) @ ( inverse_inverse @ A @ ( F3 @ X ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_4840_DERIV__compose__FDERIV,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: real > real,F6: real,G3: A > real,X: A,G6: A > real,S2: set @ A] :
( ( has_field_derivative @ real @ F3 @ F6 @ ( topolo174197925503356063within @ real @ ( G3 @ X ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( F3 @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G6 @ X4 ) @ F6 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_compose_FDERIV
thf(fact_4841_has__derivative__cos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,G6: A > real,X: A,S2: set @ A] :
( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( cos @ real @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G6 @ X4 ) @ ( uminus_uminus @ real @ ( sin @ real @ ( G3 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_cos
thf(fact_4842_has__derivative__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,F6: A > B,X: A,S3: set @ A,N: nat] :
( ( has_derivative @ A @ B @ F3 @ F6 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( has_derivative @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ^ [Y4: A] : ( times_times @ B @ ( times_times @ B @ ( semiring_1_of_nat @ B @ N ) @ ( F6 @ Y4 ) ) @ ( power_power @ B @ ( F3 @ X ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_power
thf(fact_4843_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X: A,G6: A > real,S2: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( ln_ln @ real @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G6 @ X4 ) @ ( inverse_inverse @ real @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivative_ln
thf(fact_4844_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: C > A,F6: C > A,X: C,S3: set @ C,G3: C > A,G6: C > A] :
( ( has_derivative @ C @ A @ F3 @ F6 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( has_derivative @ C @ A @ G3 @ G6 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( ( G3 @ X )
!= ( 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 @ X ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G3 @ X ) ) @ ( G6 @ H2 ) ) @ ( inverse_inverse @ A @ ( G3 @ X ) ) ) ) @ ( divide_divide @ A @ ( F6 @ H2 ) @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_4845_has__derivative__prod,axiom,
! [B: $tType,I6: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I5: set @ I6,F3: I6 > A > B,F6: I6 > A > B,X: A,S3: set @ A] :
( ! [I3: I6] :
( ( member @ I6 @ I3 @ I5 )
=> ( has_derivative @ A @ B @ ( F3 @ I3 ) @ ( F6 @ I3 ) @ ( topolo174197925503356063within @ A @ X @ S3 ) ) )
=> ( has_derivative @ A @ B
@ ^ [X4: A] :
( groups7121269368397514597t_prod @ I6 @ B
@ ^ [I: I6] : ( F3 @ I @ X4 )
@ I5 )
@ ^ [Y4: A] :
( groups7311177749621191930dd_sum @ I6 @ B
@ ^ [I: I6] :
( times_times @ B @ ( F6 @ I @ Y4 )
@ ( groups7121269368397514597t_prod @ I6 @ B
@ ^ [J: I6] : ( F3 @ J @ X )
@ ( minus_minus @ ( set @ I6 ) @ I5 @ ( insert @ I6 @ I @ ( bot_bot @ ( set @ I6 ) ) ) ) ) )
@ I5 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_prod
thf(fact_4846_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,G6: A > real,X: A,X8: set @ A,F3: A > real,F6: A > real] :
( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ X8 ) )
=> ( ( has_derivative @ A @ real @ F3 @ F6 @ ( topolo174197925503356063within @ A @ X @ X8 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( ( member @ A @ X @ X8 )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( powr @ real @ ( G3 @ X4 ) @ ( F3 @ X4 ) )
@ ^ [H2: A] : ( times_times @ real @ ( powr @ real @ ( G3 @ X ) @ ( F3 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F6 @ H2 ) @ ( ln_ln @ real @ ( G3 @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G6 @ H2 ) @ ( F3 @ X ) ) @ ( G3 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ X8 ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_4847_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X: A,G6: A > real,S2: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( sqrt @ ( G3 @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G6 @ X4 ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G3 @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_4848_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X: A,G6: A > real,S2: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G3 @ X ) )
=> ( ( ord_less @ real @ ( G3 @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( 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 @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_4849_has__derivative__floor,axiom,
! [Aa: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( archim2362893244070406136eiling @ Aa )
& ( topolo2564578578187576103pology @ Aa ) )
=> ! [G3: A > real,X: A,F3: real > Aa,G6: A > real,S2: set @ A] :
( ( topolo3448309680560233919inuous @ real @ Aa @ ( topolo174197925503356063within @ real @ ( G3 @ X ) @ ( top_top @ ( set @ real ) ) ) @ F3 )
=> ( ~ ( member @ Aa @ ( F3 @ ( G3 @ X ) ) @ ( ring_1_Ints @ Aa ) )
=> ( ( has_derivative @ A @ real @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ Aa @ ( F3 @ ( G3 @ X4 ) ) ) )
@ ^ [X4: A] : ( times_times @ real @ ( G6 @ X4 ) @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% has_derivative_floor
thf(fact_4850_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K5: A,X: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C3 ) @ N5 ) @ ( power_power @ A @ K5 @ N5 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( filterlim @ A @ A
@ ^ [H2: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X @ H2 ) @ N5 ) @ ( power_power @ A @ X @ N5 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N5 ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_4851_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F3: B > A,L: A,F5: filter @ B] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C3 ) )
@ F5 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_4852_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F3: B > A,L: A,F5: filter @ B] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C3 @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C3 @ L ) )
@ F5 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_4853_power__tendsto__0__iff,axiom,
! [A: $tType,N: nat,F3: A > real,F5: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real
@ ^ [X4: A] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ).
% power_tendsto_0_iff
thf(fact_4854_tendsto__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: A > A,A3: A,F5: filter @ A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( ( cos @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X4: A] : ( tan @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_tan
thf(fact_4855_tendsto__mult,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F3: B > A,A3: A,F5: filter @ B,G3: B > A,B2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ A3 @ B2 ) )
@ F5 ) ) ) ) ).
% tendsto_mult
thf(fact_4856_tendsto__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F3: B > A,L: A,F5: filter @ B,C3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C3 @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C3 @ L ) )
@ F5 ) ) ) ).
% tendsto_mult_left
thf(fact_4857_tendsto__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F3: B > A,L: A,F5: filter @ B,C3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C3 ) )
@ F5 ) ) ) ).
% tendsto_mult_right
thf(fact_4858_tendsto__mult__one,axiom,
! [B: $tType,D: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F3: D > B,F5: filter @ D,G3: D > B] :
( ( filterlim @ D @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F5 )
=> ( ( filterlim @ D @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F5 )
=> ( filterlim @ D @ B
@ ^ [X4: D] : ( times_times @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) )
@ F5 ) ) ) ) ).
% tendsto_mult_one
thf(fact_4859_continuous__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F5: filter @ B,F3: B > A,C3: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F5 @ F3 )
=> ( topolo3448309680560233919inuous @ B @ A @ F5
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C3 ) ) ) ) ).
% continuous_mult_right
thf(fact_4860_continuous__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F5: filter @ B,F3: B > A,C3: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F5 @ F3 )
=> ( topolo3448309680560233919inuous @ B @ A @ F5
@ ^ [X4: B] : ( times_times @ A @ C3 @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_mult_left
thf(fact_4861_continuous__mult_H,axiom,
! [B: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [F5: filter @ D,F3: D > B,G3: D > B] :
( ( topolo3448309680560233919inuous @ D @ B @ F5 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ B @ F5 @ G3 )
=> ( topolo3448309680560233919inuous @ D @ B @ F5
@ ^ [X4: D] : ( times_times @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ).
% continuous_mult'
thf(fact_4862_continuous__mult,axiom,
! [A: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F5: filter @ D,F3: D > A,G3: D > A] :
( ( topolo3448309680560233919inuous @ D @ A @ F5 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ A @ F5 @ G3 )
=> ( topolo3448309680560233919inuous @ D @ A @ F5
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ).
% continuous_mult
thf(fact_4863_tendsto__divide,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,A3: A,F5: filter @ B,G3: B > A,B2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A3 @ B2 ) )
@ F5 ) ) ) ) ) ).
% tendsto_divide
thf(fact_4864_tendsto__divide__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,F5: filter @ B,C3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( divide_divide @ A @ ( F3 @ X4 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_divide_zero
thf(fact_4865_tendsto__mult__right__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F5: filter @ D,C3: A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X4: D] : ( times_times @ A @ C3 @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_4866_tendsto__mult__left__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F5: filter @ D,C3: A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_4867_tendsto__mult__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F5: filter @ D,G3: D > A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( ( filterlim @ D @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_4868_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ).
% LIM_zero
thf(fact_4869_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 )
= ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ).
% LIM_zero_iff
thf(fact_4870_Lim__transform,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: B > A,A3: A,F5: filter @ B,F3: B > A] :
( ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 ) ) ) ) ).
% Lim_transform
thf(fact_4871_Lim__transform2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,A3: A,F5: filter @ B,G3: B > A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 ) ) ) ) ).
% Lim_transform2
thf(fact_4872_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ).
% LIM_zero_cancel
thf(fact_4873_Lim__transform__eq,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,G3: B > A,F5: filter @ B,A3: A] :
( ( filterlim @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
= ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 ) ) ) ) ).
% Lim_transform_eq
thf(fact_4874_tendsto__norm__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ).
% tendsto_norm_zero
thf(fact_4875_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
= ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_4876_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_4877_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I5: set @ B,F3: A > B > C,F5: filter @ A] :
( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( F3 @ X4 @ I3 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) )
=> ( filterlim @ A @ C
@ ^ [I: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F3 @ I ) @ I5 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ).
% tendsto_null_sum
thf(fact_4878_tendsto__sgn,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,L: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( ( L
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( sgn_sgn @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) )
@ F5 ) ) ) ) ).
% tendsto_sgn
thf(fact_4879_continuous__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F5: filter @ A,F3: A > B,G3: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ F5 @ G3 )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F5
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ).
% continuous_Pair
thf(fact_4880_tendsto__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,A3: B,F5: filter @ A,G3: A > C,B2: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F5 )
=> ( ( filterlim @ A @ C @ G3 @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F5 )
=> ( 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 ) )
@ F5 ) ) ) ) ).
% tendsto_Pair
thf(fact_4881_tendsto__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A3: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_inverse
thf(fact_4882_tendsto__add__zero,axiom,
! [B: $tType,D: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F3: D > B,F5: filter @ D,G3: D > B] :
( ( filterlim @ D @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( filterlim @ D @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ D @ B
@ ^ [X4: D] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% tendsto_add_zero
thf(fact_4883_tendsto__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,A3: A,F5: filter @ C] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( ( cosh @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ C @ A
@ ^ [X4: C] : ( tanh @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_tanh
thf(fact_4884_tendsto__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: A > A,A3: A,F5: filter @ A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( ( sin @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X4: A] : ( cot @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_cot
thf(fact_4885_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F3 )
= ( filterlim @ A @ B
@ ^ [H2: A] : ( F3 @ ( plus_plus @ A @ X @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ X ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_4886_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_4887_LIM__not__zero,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( zero @ Aa )
& ( topological_t2_space @ Aa ) )
=> ! [K2: Aa,A3: A] :
( ( K2
!= ( zero_zero @ Aa ) )
=> ~ ( filterlim @ A @ Aa
@ ^ [X4: A] : K2
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_4888_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_4889_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_4890_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_4891_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F3: A > B,F5: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% tendsto_null_power
thf(fact_4892_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_4893_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 )
=> ( ! [X3: A] :
( ( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F3 ) )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F3 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT2
thf(fact_4894_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 )
=> ( ! [X3: A] :
( ( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F3 ) )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F3 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT
thf(fact_4895_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > A,A3: A,D5: A] :
( ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ A3 @ H2 ) ) @ ( F3 @ A3 ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( 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 @ D5 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_4896_continuous__at__within__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,S2: set @ A,F3: A > B,G3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ G3 )
=> ( ( ( G3 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 )
@ ^ [X4: A] : ( divide_divide @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_4897_isCont__mult,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V4412858255891104859lgebra @ 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] : ( times_times @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ).
% isCont_mult
thf(fact_4898_continuous__at__within__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A3: A,S2: set @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ F3 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 )
@ ^ [X4: A] : ( inverse_inverse @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_4899_continuous__at__within__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,S2: set @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ F3 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 )
@ ^ [X4: A] : ( sgn_sgn @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_at_within_sgn
thf(fact_4900_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H2 ) ) @ ( F3 @ X ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_4901_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A] :
( ( has_field_derivative @ A @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H2 ) ) @ ( F3 @ X ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_4902_lim__exp__minus__1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( filterlim @ A @ A
@ ^ [Z4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z4 ) @ ( one_one @ A ) ) @ Z4 )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% lim_exp_minus_1
thf(fact_4903_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K2: real,F3: A > B,K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K2 )
=> ( ! [H4: A] :
( ( H4
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H4 ) @ K2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ H4 ) ) @ ( times_times @ real @ K5 @ ( real_V7770717601297561774m_norm @ A @ H4 ) ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_4904_isCont__eq__Lb,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( ord_less_eq @ A @ M8 @ ( F3 @ X5 ) ) )
& ? [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ( F3 @ X3 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_4905_isCont__eq__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ M8 ) )
& ? [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ( F3 @ X3 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_4906_isCont__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ M8 ) ) ) ) ) ).
% isCont_bounded
thf(fact_4907_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D5: A,X: A] :
( ( has_derivative @ A @ A @ F3 @ ( times_times @ A @ D5 ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H2 ) ) @ ( F3 @ X ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_4908_isCont__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ 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 )
=> ( ( ( G3 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( divide_divide @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ) ).
% isCont_divide
thf(fact_4909_isCont__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( sgn_sgn @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% isCont_sgn
thf(fact_4910_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > B,F5: filter @ B,A3: A] :
( ( filterlim @ A @ B @ F3 @ F5 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X4: A] : ( F3 @ ( plus_plus @ A @ X4 @ A3 ) )
@ F5
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_4911_continuous__within__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S2: set @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S2 ) @ F3 )
=> ( ( ( cos @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S2 )
@ ^ [X4: A] : ( tan @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_within_tan
thf(fact_4912_continuous__within__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S2: set @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S2 ) @ F3 )
=> ( ( ( sin @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S2 )
@ ^ [X4: A] : ( cot @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_within_cot
thf(fact_4913_continuous__at__within__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: C,A6: set @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A6 ) @ F3 )
=> ( ( ( cosh @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A6 )
@ ^ [X4: C] : ( tanh @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_at_within_tanh
thf(fact_4914_CARAT__DERIV,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,L: A,X: A] :
( ( has_field_derivative @ A @ F3 @ L @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ? [G4: A > A] :
( ! [Z4: A] :
( ( minus_minus @ A @ ( F3 @ Z4 ) @ ( F3 @ X ) )
= ( times_times @ A @ ( G4 @ Z4 ) @ ( minus_minus @ A @ Z4 @ X ) ) )
& ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ G4 )
& ( ( G4 @ X )
= L ) ) ) ) ) ).
% CARAT_DERIV
thf(fact_4915_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
( ! [X5: real] :
( ( ( ord_less_eq @ real @ A3 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ M8 ) )
& ! [N8: A] :
( ( ord_less @ A @ N8 @ M8 )
=> ? [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ord_less @ A @ N8 @ ( F3 @ X3 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_4916_isCont__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( tan @ A ) ) ) ) ).
% isCont_tan
thf(fact_4917_isCont__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( cot @ A ) ) ) ) ).
% isCont_cot
thf(fact_4918_isCont__tanh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cosh @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( tanh @ A ) ) ) ) ).
% isCont_tanh
thf(fact_4919_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: real,A3: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S2 )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S2 )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( A3 @ N5 ) @ ( power_power @ A @ X3 @ N5 ) )
@ ( F3 @ X3 ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( A3 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0
thf(fact_4920_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: real,A3: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S2 )
=> ( ! [X3: A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S2 )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( A3 @ N5 ) @ ( power_power @ A @ X3 @ N5 ) )
@ ( F3 @ X3 ) ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( A3 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0_strong
thf(fact_4921_lemma__termdiff5,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_Vector_banach @ B ) )
=> ! [K2: real,F3: nat > real,G3: A > nat > B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K2 )
=> ( ( summable @ real @ F3 )
=> ( ! [H4: A,N3: nat] :
( ( H4
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H4 ) @ K2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G3 @ H4 @ N3 ) ) @ ( times_times @ real @ ( F3 @ N3 ) @ ( real_V7770717601297561774m_norm @ A @ H4 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H2: A] : ( suminf @ B @ ( G3 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_4922_isCont__tan_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( cos @ A @ ( F3 @ A3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( tan @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% isCont_tan'
thf(fact_4923_isCont__cot_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( sin @ A @ ( F3 @ A3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( cot @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% isCont_cot'
thf(fact_4924_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
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ W3 @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% isCont_polynom
thf(fact_4925_isCont__powser__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,X: A] :
( ! [Y3: A] :
( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ Y3 @ N5 ) ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ X4 @ N5 ) ) ) ) ) ) ).
% isCont_powser_converges_everywhere
thf(fact_4926_GMVT_H,axiom,
! [A3: real,B2: real,F3: real > real,G3: real > real,G6: real > real,F6: real > real] :
( ( ord_less @ real @ A3 @ B2 )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ A3 @ Z )
=> ( ( ord_less_eq @ real @ Z @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) @ F3 ) ) )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ A3 @ Z )
=> ( ( ord_less_eq @ real @ Z @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) @ G3 ) ) )
=> ( ! [Z: real] :
( ( ord_less @ real @ A3 @ Z )
=> ( ( ord_less @ real @ Z @ B2 )
=> ( has_field_derivative @ real @ G3 @ ( G6 @ Z ) @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ! [Z: real] :
( ( ord_less @ real @ A3 @ Z )
=> ( ( ord_less @ real @ Z @ B2 )
=> ( has_field_derivative @ real @ F3 @ ( F6 @ Z ) @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [C2: real] :
( ( ord_less @ real @ A3 @ C2 )
& ( ord_less @ real @ C2 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A3 ) ) @ ( G6 @ C2 ) )
= ( times_times @ real @ ( minus_minus @ real @ ( G3 @ B2 ) @ ( G3 @ A3 ) ) @ ( F6 @ C2 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_4927_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K5: A,X: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ K5 @ N5 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C3 @ N5 ) @ ( power_power @ A @ X4 @ N5 ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_4928_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
@ ^ [N5: nat] : ( times_times @ Aa @ ( C3 @ N5 ) @ ( power_power @ Aa @ K5 @ N5 ) ) )
=> ( ( 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
@ ^ [N5: nat] : ( times_times @ Aa @ ( C3 @ N5 ) @ ( power_power @ Aa @ ( F3 @ X4 ) @ N5 ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_4929_summable__Leibniz_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( ( ord_less @ real @ ( A3 @ ( zero_zero @ nat ) ) @ ( zero_zero @ real ) )
=> ! [N4: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_4930_summable__Leibniz_I2_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( A3 @ ( zero_zero @ nat ) ) )
=> ! [N4: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_4931_summable__Leibniz_H_I4_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( ord_less_eq @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_4932_tendsto__zero__mult__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N5: nat] : ( times_times @ A @ ( A3 @ N5 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_4933_tendsto__zero__mult__left__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N5: nat] : ( times_times @ A @ C3 @ ( A3 @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_4934_tendsto__zero__divide__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( A3 @ N5 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_4935_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
@ ^ [N5: nat] : ( F3 @ ( suc @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Suc
thf(fact_4936_LIMSEQ__imp__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N5: nat] : ( F3 @ ( suc @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_imp_Suc
thf(fact_4937_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X: A,A3: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ A3 ) )
=> ( ord_less_eq @ A @ X @ A3 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_4938_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X: A,A3: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ A3 @ ( X8 @ N3 ) ) )
=> ( ord_less_eq @ A @ A3 @ X ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_4939_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,L: A,N7: nat,C5: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ A @ C5 @ ( F3 @ N3 ) ) )
=> ( ord_less_eq @ A @ C5 @ L ) ) ) ) ).
% Lim_bounded2
thf(fact_4940_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,L: A,M5: nat,C5: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ A @ ( F3 @ N3 ) @ C5 ) )
=> ( ord_less_eq @ A @ L @ C5 ) ) ) ) ).
% Lim_bounded
thf(fact_4941_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,X: A,Y7: nat > A,Y: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_4942_lim__mono,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [N7: nat,X8: nat > A,Y7: nat > A,X: A,Y: A] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% lim_mono
thf(fact_4943_summable__LIMSEQ__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ_zero
thf(fact_4944_incseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,L5: A,N: nat] :
( ( order_mono @ nat @ A @ X8 )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( X8 @ N ) @ L5 ) ) ) ) ).
% incseq_le
thf(fact_4945_mult__nat__left__at__top,axiom,
! [C3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( filterlim @ nat @ nat @ ( times_times @ nat @ C3 ) @ ( at_top @ nat ) @ ( at_top @ nat ) ) ) ).
% mult_nat_left_at_top
thf(fact_4946_mult__nat__right__at__top,axiom,
! [C3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( filterlim @ nat @ nat
@ ^ [X4: nat] : ( times_times @ nat @ X4 @ C3 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_4947_monoseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: nat > A,X: A] :
( ( topological_monoseq @ A @ A3 )
=> ( ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ! [N4: nat] : ( ord_less_eq @ A @ ( A3 @ N4 ) @ X )
& ! [M3: nat,N4: nat] :
( ( ord_less_eq @ nat @ M3 @ N4 )
=> ( ord_less_eq @ A @ ( A3 @ M3 ) @ ( A3 @ N4 ) ) ) )
| ( ! [N4: nat] : ( ord_less_eq @ A @ X @ ( A3 @ N4 ) )
& ! [M3: nat,N4: nat] :
( ( ord_less_eq @ nat @ M3 @ N4 )
=> ( ord_less_eq @ A @ ( A3 @ N4 ) @ ( A3 @ M3 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_4948_lim__const__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A] :
( filterlim @ nat @ A
@ ^ [N5: nat] : ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_const_over_n
thf(fact_4949_lim__inverse__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_inverse_n
thf(fact_4950_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X8: nat > A,X: A,L: nat] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( X8 @ ( times_times @ nat @ N5 @ L ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_4951_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
@ ^ [N5: nat] : ( minus_minus @ A @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) ) ) ) ) ).
% telescope_summable'
thf(fact_4952_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
@ ^ [N5: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N5 ) ) @ ( F3 @ N5 ) ) ) ) ) ).
% telescope_summable
thf(fact_4953_nested__sequence__unique,axiom,
! [F3: nat > real,G3: nat > real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( G3 @ ( suc @ N3 ) ) @ ( G3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ ( G3 @ N3 ) )
=> ( ( filterlim @ nat @ real
@ ^ [N5: nat] : ( minus_minus @ real @ ( F3 @ N5 ) @ ( G3 @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) )
=> ? [L4: real] :
( ! [N4: nat] : ( ord_less_eq @ real @ ( F3 @ N4 ) @ L4 )
& ( filterlim @ nat @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) )
& ! [N4: nat] : ( ord_less_eq @ real @ L4 @ ( G3 @ N4 ) )
& ( filterlim @ nat @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_4954_LIMSEQ__inverse__zero,axiom,
! [X8: nat > real] :
( ! [R4: real] :
? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less @ real @ R4 @ ( X8 @ N3 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] : ( inverse_inverse @ real @ ( X8 @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_4955_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim @ nat @ real
@ ^ [N5: nat] : ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_4956_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R3: real] :
( filterlim @ nat @ real
@ ^ [N5: nat] : ( plus_plus @ real @ R3 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R3 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_4957_increasing__LIMSEQ,axiom,
! [F3: nat > real,L: real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ L )
=> ( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [N4: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F3 @ N4 ) @ E2 ) ) )
=> ( filterlim @ nat @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_4958_lim__1__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_1_over_n
thf(fact_4959_LIMSEQ__n__over__Suc__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N5 ) @ ( semiring_1_of_nat @ A @ ( suc @ N5 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_4960_LIMSEQ__Suc__n__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ N5 ) ) @ ( semiring_1_of_nat @ A @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_4961_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
@ ^ [N5: nat] : ( minus_minus @ A @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) )
@ ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ C3 ) ) ) ) ).
% telescope_sums'
thf(fact_4962_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
@ ^ [N5: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N5 ) ) @ ( F3 @ N5 ) )
@ ( minus_minus @ A @ C3 @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_4963_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F4: nat > A,S6: A] :
( filterlim @ nat @ A
@ ^ [N5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S6 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_4964_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R3: real] :
( filterlim @ nat @ real
@ ^ [N5: nat] : ( plus_plus @ real @ R3 @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R3 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_4965_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L5: A,R3: real] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ No @ N4 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N4 ) @ L5 ) ) @ R3 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_4966_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L5: A] :
( ! [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
=> ? [No2: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No2 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N3 ) @ L5 ) ) @ R4 ) ) )
=> ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_4967_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [No3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ No3 @ N5 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N5 ) @ L5 ) ) @ R ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_4968_LIMSEQ__power__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_power_zero
thf(fact_4969_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: B > nat,F5: filter @ B,X: A] :
( ( filterlim @ B @ nat @ F3 @ ( at_top @ nat ) @ F5 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y4: B] : ( power_power @ A @ X @ ( F3 @ Y4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_power_zero
thf(fact_4970_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R3: real] :
( filterlim @ nat @ real
@ ^ [N5: nat] : ( times_times @ real @ R3 @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R3 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_4971_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_4972_summable__Leibniz_I1_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( summable @ real
@ ^ [N5: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N5 ) @ ( A3 @ N5 ) ) ) ) ) ).
% summable_Leibniz(1)
thf(fact_4973_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Df: A,Z3: A,S2: nat > A,A3: A] :
( ( has_field_derivative @ A @ F3 @ Df @ ( topolo174197925503356063within @ A @ Z3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( S2 @ N3 )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ Z3 @ ( S2 @ N5 ) ) ) @ ( F3 @ Z3 ) ) @ ( S2 @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) )
=> ( Df = A3 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_4974_powser__times__n__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N5 ) @ ( power_power @ A @ X @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_4975_lim__n__over__pown,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N5 ) @ ( power_power @ A @ X @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_4976_summable,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( summable @ real
@ ^ [N5: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N5 ) @ ( A3 @ N5 ) ) ) ) ) ) ).
% summable
thf(fact_4977_cos__diff__limit__1,axiom,
! [Theta: nat > real,Theta2: real] :
( ( filterlim @ nat @ real
@ ^ [J: nat] : ( cos @ real @ ( minus_minus @ real @ ( Theta @ J ) @ Theta2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ~ ! [K: nat > int] :
~ ( filterlim @ nat @ real
@ ^ [J: nat] : ( minus_minus @ real @ ( Theta @ J ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K @ J ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ Theta2 )
@ ( at_top @ nat ) ) ) ).
% cos_diff_limit_1
thf(fact_4978_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim @ nat @ real
@ ^ [J: nat] : ( cos @ real @ ( Theta @ J ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ? [K: nat > int] :
( filterlim @ nat @ real
@ ^ [J: nat] : ( minus_minus @ real @ ( Theta @ J ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K @ J ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% cos_limit_1
thf(fact_4979_summable__Leibniz_I4_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(4)
thf(fact_4980_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_4981_summable__Leibniz_H_I2_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_4982_summable__Leibniz_H_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_4983_sums__alternating__upper__lower,axiom,
! [A3: nat > real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ? [L4: real] :
( ! [N4: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ L4 )
& ( filterlim @ nat @ real
@ ^ [N5: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) )
& ! [N4: nat] :
( ord_less_eq @ real @ L4
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N5: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_4984_summable__Leibniz_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(5)
thf(fact_4985_summable__Leibniz_H_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N3 ) ) @ ( A3 @ N3 ) )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A3 @ I ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_4986_filterlim__sequentially__Suc,axiom,
! [A: $tType,F3: nat > A,F5: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X4: nat] : ( F3 @ ( suc @ X4 ) )
@ F5
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F3 @ F5 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_4987_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F6: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ F6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ( filterlim @ A @ B
@ ^ [Y4: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y4 @ X ) ) ) @ ( minus_minus @ B @ ( F3 @ Y4 ) @ ( plus_plus @ B @ ( F3 @ X ) @ ( F6 @ ( minus_minus @ A @ Y4 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_4988_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,D5: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ D5 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D5 )
& ( filterlim @ A @ real
@ ^ [H2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ ( plus_plus @ A @ X @ H2 ) ) @ ( F3 @ X ) ) @ ( D5 @ 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_4989_bounded__linear__mult__left,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [Y: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ Y ) ) ) ).
% bounded_linear_mult_left
thf(fact_4990_bounded__linear__const__mult,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G3: C > A,X: A] :
( ( real_V3181309239436604168linear @ C @ A @ G3 )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X4: C] : ( times_times @ A @ X @ ( G3 @ X4 ) ) ) ) ) ).
% bounded_linear_const_mult
thf(fact_4991_bounded__linear__mult__const,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G3: C > A,Y: A] :
( ( real_V3181309239436604168linear @ C @ A @ G3 )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X4: C] : ( times_times @ A @ ( G3 @ X4 ) @ Y ) ) ) ) ).
% bounded_linear_mult_const
thf(fact_4992_bounded__linear__mult__right,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A] : ( real_V3181309239436604168linear @ A @ A @ ( times_times @ A @ X ) ) ) ).
% bounded_linear_mult_right
thf(fact_4993_real__bounded__linear,axiom,
( ( real_V3181309239436604168linear @ real @ real )
= ( ^ [F4: real > real] :
? [C4: real] :
( F4
= ( ^ [X4: real] : ( times_times @ real @ X4 @ C4 ) ) ) ) ) ).
% real_bounded_linear
thf(fact_4994_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X4: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_4995_bounded__linear_Obounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K9: real] :
! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K9 ) ) ) ) ).
% bounded_linear.bounded
thf(fact_4996_bounded__linear_Otendsto__zero,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G3: C > A,F5: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( filterlim @ C @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( F3 @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_4997_bounded__linear_Ononneg__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K9: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ K9 )
& ! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K9 ) ) ) ) ) ).
% bounded_linear.nonneg_bounded
thf(fact_4998_bounded__linear_Opos__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K9 ) ) ) ) ) ).
% bounded_linear.pos_bounded
thf(fact_4999_bounded__linear__intro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,K5: real] :
( ! [X3: A,Y3: A] :
( ( F3 @ ( plus_plus @ A @ X3 @ Y3 ) )
= ( plus_plus @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ! [R4: real,X3: A] :
( ( F3 @ ( real_V8093663219630862766scaleR @ A @ R4 @ X3 ) )
= ( real_V8093663219630862766scaleR @ B @ R4 @ ( F3 @ X3 ) ) )
=> ( ! [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X3 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K5 ) )
=> ( real_V3181309239436604168linear @ A @ B @ F3 ) ) ) ) ) ).
% bounded_linear_intro
thf(fact_5000_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_5001_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F6: A > B,X: A,F3: A > B,S2: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F6 )
=> ( ( filterlim @ A @ B
@ ^ [Y4: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y4 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y4 ) @ ( F3 @ X ) ) @ ( F6 @ ( minus_minus @ A @ Y4 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ B @ F3 @ F6 @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivativeI
thf(fact_5002_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F6: A > B,X: A,S2: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ( filterlim @ A @ B
@ ^ [Y4: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y4 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y4 ) @ ( F3 @ X ) ) @ ( F6 @ ( minus_minus @ A @ Y4 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_5003_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F6: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ F6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ? [E4: A > B] :
( ! [H2: A] :
( ( F3 @ ( plus_plus @ A @ X @ H2 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X ) @ ( F6 @ 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_5004_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F6: A > B,X: A,S2: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ( filterlim @ A @ B
@ ^ [Y4: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y4 @ X ) ) ) @ ( minus_minus @ B @ ( F3 @ Y4 ) @ ( plus_plus @ B @ ( F3 @ X ) @ ( F6 @ ( minus_minus @ A @ Y4 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% has_derivative_within
thf(fact_5005_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F4: A > B,F7: A > B,F8: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ B
@ ^ [Y4: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y4
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X4: A] : X4 ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F4 @ Y4 )
@ ( F4
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X4: A] : X4 ) ) )
@ ( F7
@ ( minus_minus @ A @ Y4
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X4: A] : X4 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F8 ) ) ) ) ) ).
% has_derivative_def
thf(fact_5006_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X: A,S3: set @ A,F3: A > B,F6: A > B] :
( ( member @ A @ X @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( has_derivative @ A @ B @ F3 @ F6 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ? [E4: A > B] :
( ! [H2: A] :
( ( member @ A @ ( plus_plus @ A @ X @ H2 ) @ S3 )
=> ( ( F3 @ ( plus_plus @ A @ X @ H2 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X ) @ ( F6 @ 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_5007_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,L: A] :
( ( filterlim @ A @ A
@ ^ [X4: A] : ( F3 @ ( divide_divide @ A @ ( one_one @ A ) @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_5008_lim__explicit,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,F0: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ F0 ) @ ( at_top @ nat ) )
= ( ! [S7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S7 )
=> ( ( member @ A @ F0 @ S7 )
=> ? [N6: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ N6 @ N5 )
=> ( member @ A @ ( F3 @ N5 ) @ S7 ) ) ) ) ) ) ) ).
% lim_explicit
thf(fact_5009_continuous__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F5: filter @ A,F3: A > B,G3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ G3 )
=> ( ( ( G3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X4: A] : X4 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X4: A] : ( divide_divide @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_5010_continuous__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F5: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X4: A] : X4 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X4: A] : ( inverse_inverse @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_inverse
thf(fact_5011_continuous__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F5: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X4: A] : X4 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X4: A] : ( sgn_sgn @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_sgn
thf(fact_5012_tendsto__inverse__0,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ( filterlim @ A @ A @ ( inverse_inverse @ A ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_infinity @ A ) ) ) ).
% tendsto_inverse_0
thf(fact_5013_tendsto__mult__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,C3: A,F5: filter @ B,G3: B > A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F5 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A @ G3 @ ( at_infinity @ A ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_infinity @ A )
@ F5 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_5014_tendsto__divide__0,axiom,
! [A: $tType,C: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: C > A,C3: A,F5: filter @ C,G3: C > A] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F5 )
=> ( ( filterlim @ C @ A @ G3 @ ( at_infinity @ A ) @ F5 )
=> ( filterlim @ C @ A
@ ^ [X4: C] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_divide_0
thf(fact_5015_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F3: A > B,F5: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ( at_infinity @ B )
@ F5 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_5016_continuous__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F5 @ F3 )
=> ( ( ( cos @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X4: A] : X4 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F5
@ ^ [X4: A] : ( tan @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_tan
thf(fact_5017_continuous__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F5 @ F3 )
=> ( ( ( sin @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X4: A] : X4 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F5
@ ^ [X4: A] : ( cot @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_cot
thf(fact_5018_continuous__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F5 @ F3 )
=> ( ( ( cosh @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ C @ C @ F5
@ ^ [X4: C] : X4 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ F5
@ ^ [X4: C] : ( tanh @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_tanh
thf(fact_5019_filterlim__inverse__at__infinity,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ( filterlim @ A @ A @ ( inverse_inverse @ A ) @ ( at_infinity @ A ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% filterlim_inverse_at_infinity
thf(fact_5020_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X4: A] : ( inverse_inverse @ B @ ( G3 @ X4 ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F5 )
= ( filterlim @ A @ B @ G3 @ ( at_infinity @ B ) @ F5 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_5021_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_5022_filterlim__divide__at__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,C3: A,F5: filter @ A,G3: A > A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F5 )
=> ( ( filterlim @ A @ A @ G3 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F5 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X4: A] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_infinity @ A )
@ F5 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_5023_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C3: nat > A,K2: nat,N: nat,B6: real] :
( ( ( C3 @ K2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K2 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( eventually @ A
@ ^ [Z4: A] :
( ord_less_eq @ real @ B6
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C3 @ I ) @ ( power_power @ A @ Z4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_5024_tendsto__exp__limit__at__right,axiom,
! [X: real] :
( filterlim @ real @ real
@ ^ [Y4: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ X @ Y4 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ Y4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% tendsto_exp_limit_at_right
thf(fact_5025_GMVT,axiom,
! [A3: real,B2: real,F3: real > real,G3: real > real] :
( ( ord_less @ real @ A3 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ( ! [X3: real] :
( ( ( ord_less @ real @ A3 @ X3 )
& ( ord_less @ real @ X3 @ B2 ) )
=> ( differentiable @ real @ real @ F3 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ G3 ) )
=> ( ! [X3: real] :
( ( ( ord_less @ real @ A3 @ X3 )
& ( ord_less @ real @ X3 @ B2 ) )
=> ( differentiable @ real @ real @ G3 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [G_c: real,F_c: real,C2: real] :
( ( has_field_derivative @ real @ G3 @ G_c @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
& ( has_field_derivative @ real @ F3 @ F_c @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ A3 @ C2 )
& ( ord_less @ real @ C2 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A3 ) ) @ G_c )
= ( times_times @ real @ ( minus_minus @ real @ ( G3 @ B2 ) @ ( G3 @ A3 ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_5026_greaterThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_greaterThan @ A @ X )
= ( set_ord_greaterThan @ A @ Y ) )
= ( X = Y ) ) ) ).
% greaterThan_eq_iff
thf(fact_5027_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K2: A] :
( ( member @ A @ I2 @ ( set_ord_greaterThan @ A @ K2 ) )
= ( ord_less @ A @ K2 @ I2 ) ) ) ).
% greaterThan_iff
thf(fact_5028_eventually__sequentially__Suc,axiom,
! [P2: nat > $o] :
( ( eventually @ nat
@ ^ [I: nat] : ( P2 @ ( suc @ I ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P2 @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_5029_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X ) @ ( set_ord_greaterThan @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% greaterThan_subset_iff
thf(fact_5030_Compl__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K2: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K2 ) )
= ( set_ord_atMost @ A @ K2 ) ) ) ).
% Compl_greaterThan
thf(fact_5031_Compl__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K2: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K2 ) )
= ( set_ord_greaterThan @ A @ K2 ) ) ) ).
% Compl_atMost
thf(fact_5032_image__uminus__lessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_lessThan @ A @ X ) )
= ( set_ord_greaterThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_lessThan
thf(fact_5033_image__uminus__greaterThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_greaterThan @ A @ X ) )
= ( set_ord_lessThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThan
thf(fact_5034_differentiable__cmult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: A,Q2: B > A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ C3 @ ( Q2 @ T3 ) )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q2 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_left_iff
thf(fact_5035_differentiable__cmult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Q2: B > A,C3: A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ ( Q2 @ T3 ) @ C3 )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q2 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_right_iff
thf(fact_5036_greaterThan__non__empty,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
( ( set_ord_greaterThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% greaterThan_non_empty
thf(fact_5037_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ).
% greaterThan_def
thf(fact_5038_infinite__Ioi,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_greaterThan @ A @ A3 ) ) ) ).
% infinite_Ioi
thf(fact_5039_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_top @ A ) )
= ( ? [N6: A] :
! [N5: A] :
( ( ord_less_eq @ A @ N6 @ N5 )
=> ( P2 @ N5 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_5040_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,P2: A > $o] :
( ! [X3: A] :
( ( ord_less_eq @ A @ C3 @ X3 )
=> ( P2 @ X3 ) )
=> ( eventually @ A @ P2 @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_5041_eventually__sequentially,axiom,
! [P2: nat > $o] :
( ( eventually @ nat @ P2 @ ( at_top @ nat ) )
= ( ? [N6: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ N6 @ N5 )
=> ( P2 @ N5 ) ) ) ) ).
% eventually_sequentially
thf(fact_5042_eventually__sequentiallyI,axiom,
! [C3: nat,P2: nat > $o] :
( ! [X3: nat] :
( ( ord_less_eq @ nat @ C3 @ X3 )
=> ( P2 @ X3 ) )
=> ( eventually @ nat @ P2 @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_5043_lessThan__Int__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( set_ord_greaterThan @ A @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% lessThan_Int_lessThan
thf(fact_5044_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_5045_le__sequentially,axiom,
! [F5: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F5 @ ( at_top @ nat ) )
= ( ! [N6: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N6 ) @ F5 ) ) ) ).
% le_sequentially
thf(fact_5046_differentiable__mult,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F3: A > B,X: A,S2: set @ A,G3: A > B] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( differentiable @ A @ B @ G3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( times_times @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% differentiable_mult
thf(fact_5047_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P2: B > $o,G3: B > A] :
( ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P2 @ X3 )
=> ( ( F3 @ ( G3 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P2 @ X3 )
=> ( Q @ ( G3 @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( at_top @ A ) )
=> ( ( eventually @ B @ P2 @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_5048_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_5049_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F5: filter @ B,F3: B > A,X: A,G3: B > A,Y: A] :
( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
=> ( ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ A @ ( G3 @ X4 ) @ ( F3 @ X4 ) )
@ F5 )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ) ).
% tendsto_le
thf(fact_5050_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X: A,F5: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
=> ( ( eventually @ B
@ ^ [I: B] : ( ord_less_eq @ A @ A3 @ ( F3 @ I ) )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A3 @ X ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_5051_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X: A,F5: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
=> ( ( eventually @ B
@ ^ [I: B] : ( ord_less_eq @ A @ ( F3 @ I ) @ A3 )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X @ A3 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_5052_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
@ ^ [N5: B] : ( ord_less_eq @ A @ ( F3 @ N5 ) @ ( G3 @ N5 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N5: B] : ( ord_less_eq @ A @ ( G3 @ N5 ) @ ( H @ N5 ) )
@ 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_5053_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_5054_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z7: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z7 @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top
thf(fact_5055_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z7: B] :
( ( ord_less_eq @ B @ C3 @ Z7 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z7 @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_5056_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,F5: filter @ B,G3: B > A] :
( ( filterlim @ B @ A @ F3 @ ( at_top @ A ) @ F5 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ F5 )
=> ( filterlim @ B @ A @ G3 @ ( at_top @ A ) @ F5 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_5057_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_5058_greaterThanLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L2 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% greaterThanLessThan_def
thf(fact_5059_greaterThanLessThan__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [A7: A,B5: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A7 ) @ ( set_ord_lessThan @ A @ B5 ) ) ) ) ) ).
% greaterThanLessThan_eq
thf(fact_5060_greaterThanAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L2 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% greaterThanAtMost_def
thf(fact_5061_differentiable__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S2: set @ A,G3: A > B] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( differentiable @ A @ B @ G3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( G3 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( divide_divide @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% differentiable_divide
thf(fact_5062_differentiable__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S2: set @ A] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( inverse_inverse @ B @ ( F3 @ X4 ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% differentiable_inverse
thf(fact_5063_eventually__at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P2: A > $o,A3: A] :
( ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( eventually @ A
@ ^ [X4: A] : ( P2 @ ( plus_plus @ A @ X4 @ A3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_5064_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,L: A,F5: filter @ B] :
( ( eventually @ B
@ ^ [N5: B] : ( ord_less_eq @ A @ ( F3 @ N5 ) @ L )
@ F5 )
=> ( ! [X3: A] :
( ( ord_less @ A @ X3 @ L )
=> ( eventually @ B
@ ^ [N5: B] : ( ord_less @ A @ X3 @ ( F3 @ N5 ) )
@ F5 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% increasing_tendsto
thf(fact_5065_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L: A,F3: B > A,F5: filter @ B] :
( ( eventually @ B
@ ^ [N5: B] : ( ord_less_eq @ A @ L @ ( F3 @ N5 ) )
@ F5 )
=> ( ! [X3: A] :
( ( ord_less @ A @ L @ X3 )
=> ( eventually @ B
@ ^ [N5: B] : ( ord_less @ A @ ( F3 @ N5 ) @ X3 )
@ F5 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% decreasing_tendsto
thf(fact_5066_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F5: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z7: B] :
( ( ord_less @ B @ C3 @ Z7 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z7 @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_5067_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: B > A,P: A,F13: filter @ B,C3: A,L: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ P @ ( set_ord_greaterThan @ A @ P ) ) @ F13 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( L
= ( times_times @ A @ C3 @ P ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C3 @ ( F3 @ X4 ) )
@ ( topolo174197925503356063within @ A @ L @ ( set_ord_greaterThan @ A @ L ) )
@ F13 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_5068_filterlim__at__top__at__left,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P2: B > $o,G3: B > A,A3: A] :
( ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P2 @ X3 )
=> ( ( F3 @ ( G3 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P2 @ X3 )
=> ( Q @ ( G3 @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) )
=> ( ! [B4: A] :
( ( Q @ B4 )
=> ( ord_less @ A @ B4 @ A3 ) )
=> ( ( eventually @ B @ P2 @ ( 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_5069_tendsto__0__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F5: filter @ A,G3: A > C,K5: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G3 @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) ) @ K5 ) )
@ F5 )
=> ( filterlim @ A @ C @ G3 @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F5 ) ) ) ) ).
% tendsto_0_le
thf(fact_5070_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_5071_summable__bounded__partials,axiom,
! [A: $tType] :
( ( ( real_V8037385150606011577_space @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: nat > A,G3: nat > real] :
( ( eventually @ nat
@ ^ [X02: nat] :
! [A7: nat] :
( ( ord_less_eq @ nat @ X02 @ A7 )
=> ! [B5: nat] :
( ( ord_less @ nat @ A7 @ B5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or3652927894154168847AtMost @ nat @ A7 @ B5 ) ) ) @ ( G3 @ A7 ) ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_bounded_partials
thf(fact_5072_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S3: set @ A] :
( ! [A5: A,B4: A,X3: A] :
( ( member @ A @ A5 @ S3 )
=> ( ( member @ A @ B4 @ S3 )
=> ( ( ord_less_eq @ A @ A5 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B4 )
=> ( member @ A @ X3 @ 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_5073_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G3: nat > real] :
( ( eventually @ nat
@ ^ [M6: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ M6 @ N5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M6 @ N5 ) ) ) @ ( G3 @ M6 ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_Cauchy'
thf(fact_5074_atLeast__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_atLeast @ A @ X )
= ( set_ord_atLeast @ A @ Y ) )
= ( X = Y ) ) ) ).
% atLeast_eq_iff
thf(fact_5075_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_5076_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K2: A] :
( ( member @ A @ I2 @ ( set_ord_atLeast @ A @ K2 ) )
= ( ord_less_eq @ A @ K2 @ I2 ) ) ) ).
% atLeast_iff
thf(fact_5077_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_5078_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X ) @ ( set_ord_atLeast @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% atLeast_subset_iff
thf(fact_5079_image__add__atLeast,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K2 ) @ ( set_ord_atLeast @ A @ I2 ) )
= ( set_ord_atLeast @ A @ ( plus_plus @ A @ K2 @ I2 ) ) ) ) ).
% image_add_atLeast
thf(fact_5080_Compl__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K2: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K2 ) )
= ( set_ord_atLeast @ A @ K2 ) ) ) ).
% Compl_lessThan
thf(fact_5081_Compl__atLeast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K2: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K2 ) )
= ( set_ord_lessThan @ A @ K2 ) ) ) ).
% Compl_atLeast
thf(fact_5082_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_5083_image__minus__const__atLeast,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_ord_atLeast @ A @ A3 ) )
= ( set_ord_atMost @ A @ ( minus_minus @ A @ C3 @ A3 ) ) ) ) ).
% image_minus_const_atLeast
thf(fact_5084_image__minus__const__AtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atLeast @ A @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% image_minus_const_AtMost
thf(fact_5085_image__uminus__atLeast,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X ) )
= ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeast
thf(fact_5086_image__uminus__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X ) )
= ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atMost
thf(fact_5087_Int__atLeastAtMostL2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atLeast @ A @ C3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C3 ) @ B2 ) ) ) ).
% Int_atLeastAtMostL2
thf(fact_5088_Int__atLeastAtMostR2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C3 ) @ D3 ) ) ) ).
% Int_atLeastAtMostR2
thf(fact_5089_atLeast__Suc__greaterThan,axiom,
! [K2: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K2 ) )
= ( set_ord_greaterThan @ nat @ K2 ) ) ).
% atLeast_Suc_greaterThan
thf(fact_5090_infinite__Ici,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_atLeast @ A @ A3 ) ) ) ).
% infinite_Ici
thf(fact_5091_not__Iic__eq__Ici,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H: A,L3: A] :
( ( set_ord_atMost @ A @ H )
!= ( set_ord_atLeast @ A @ L3 ) ) ) ).
% not_Iic_eq_Ici
thf(fact_5092_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less_eq @ A @ L2 ) ) ) ) ) ).
% atLeast_def
thf(fact_5093_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_5094_not__UNIV__eq__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L3: A] :
( ( top_top @ ( set @ A ) )
!= ( set_ord_atLeast @ A @ L3 ) ) ) ).
% not_UNIV_eq_Ici
thf(fact_5095_not__Ici__eq__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L3: A,L: A,H: A] :
( ( set_ord_atLeast @ A @ L3 )
!= ( set_or1337092689740270186AtMost @ A @ L @ H ) ) ) ).
% not_Ici_eq_Icc
thf(fact_5096_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ( set_ord_atLeast @ A @ X )
= ( top_top @ ( set @ A ) ) )
= ( X
= ( bot_bot @ A ) ) ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_5097_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_5098_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_5099_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_5100_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_5101_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_5102_filterlim__at__top__mult__at__top,axiom,
! [A: $tType,F3: A > real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F5 )
=> ( ( filterlim @ A @ real @ G3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_top @ real )
@ F5 ) ) ) ).
% filterlim_at_top_mult_at_top
thf(fact_5103_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X4: A] :
! [Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( P2 @ Y4 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_5104_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_5105_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_5106_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_5107_atLeastLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% atLeastLessThan_def
thf(fact_5108_atLeastAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or1337092689740270186AtMost @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% atLeastAtMost_def
thf(fact_5109_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_5110_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P4: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P4 @ X4 )
& ! [Y4: A] :
( ( P4 @ Y4 )
=> ( ord_less_eq @ A @ X4 @ Y4 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_5111_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_5112_filterlim__pow__at__top,axiom,
! [A: $tType,N: nat,F3: A > real,F5: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( at_top @ real )
@ F5 ) ) ) ).
% filterlim_pow_at_top
thf(fact_5113_atMost__Int__atLeast,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N ) @ ( set_ord_atLeast @ A @ N ) )
= ( insert @ A @ N @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atMost_Int_atLeast
thf(fact_5114_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_5115_ivl__disj__un__singleton_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_greaterThan @ A @ L ) )
= ( set_ord_atLeast @ A @ L ) ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_5116_ivl__disj__un__one_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_5117_greaterThan__Suc,axiom,
! [K2: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K2 ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K2 ) @ ( insert @ nat @ ( suc @ K2 ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_5118_atLeast__Suc,axiom,
! [K2: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K2 ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K2 ) @ ( insert @ nat @ K2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_5119_filterlim__tendsto__pos__mult__at__top,axiom,
! [A: $tType,F3: A > real,C3: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_top
thf(fact_5120_filterlim__at__top__mult__tendsto__pos,axiom,
! [A: $tType,F3: A > real,C3: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( G3 @ X4 ) @ ( F3 @ X4 ) )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_at_top_mult_tendsto_pos
thf(fact_5121_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P4: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P4 @ X4 )
& ! [Y4: A] :
( ( P4 @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_5122_filterlim__at__bot__at__right,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P2: B > $o,G3: B > A,A3: A] :
( ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P2 @ X3 )
=> ( ( F3 @ ( G3 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P2 @ X3 )
=> ( Q @ ( G3 @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( ! [B4: A] :
( ( Q @ B4 )
=> ( ord_less @ A @ A3 @ B4 ) )
=> ( ( eventually @ B @ P2 @ ( 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_5123_filterlim__pow__at__bot__even,axiom,
! [N: nat,F3: real > real,F5: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F5 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_5124_GreatestI__ex__nat,axiom,
! [P2: nat > $o,B2: nat] :
( ? [X_1: nat] : ( P2 @ X_1 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P2 @ ( order_Greatest @ nat @ P2 ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_5125_Greatest__le__nat,axiom,
! [P2: nat > $o,K2: nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( ord_less_eq @ nat @ K2 @ ( order_Greatest @ nat @ P2 ) ) ) ) ).
% Greatest_le_nat
thf(fact_5126_GreatestI__nat,axiom,
! [P2: nat > $o,K2: nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P2 @ ( order_Greatest @ nat @ P2 ) ) ) ) ).
% GreatestI_nat
thf(fact_5127_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_bot @ A ) )
= ( ? [N6: A] :
! [N5: A] :
( ( ord_less_eq @ A @ N5 @ N6 )
=> ( P2 @ N5 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_5128_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X: A] :
( ( P2 @ X )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( order_Greatest @ A @ P2 )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_5129_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X: A,Q: A > $o] :
( ( P2 @ X )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( ! [Y6: A] :
( ( P2 @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P2 ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_5130_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_5131_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z7: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z7 )
@ F5 ) ) ) ) ).
% filterlim_at_bot
thf(fact_5132_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z7: B] :
( ( ord_less_eq @ B @ Z7 @ C3 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z7 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_5133_filterlim__tendsto__pos__mult__at__bot,axiom,
! [A: $tType,F3: A > real,C3: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G3 @ ( at_bot @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_bot
thf(fact_5134_filterlim__tendsto__neg__mult__at__bot,axiom,
! [A: $tType,F3: A > real,C3: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F5 )
=> ( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ G3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_tendsto_neg_mult_at_bot
thf(fact_5135_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F5: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z7: B] :
( ( ord_less @ B @ Z7 @ C3 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z7 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_5136_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F3: real > real,F5: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F5 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_5137_Bfun__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A3: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( bfun @ B @ A
@ ^ [X4: B] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ).
% Bfun_inverse
thf(fact_5138_MVT,axiom,
! [A3: real,B2: real,F3: real > real] :
( ( ord_less @ real @ A3 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B2 ) @ F3 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( differentiable @ real @ real @ F3 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [L4: real,Z: real] :
( ( ord_less @ real @ A3 @ Z )
& ( ord_less @ real @ Z @ B2 )
& ( has_field_derivative @ real @ F3 @ L4 @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) )
& ( ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A3 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A3 ) @ L4 ) ) ) ) ) ) ).
% MVT
thf(fact_5139_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq: A > A > $o,P4: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P4 @ X4 )
& ! [Y4: A] :
( ( P4 @ Y4 )
=> ( Less_eq @ X4 @ Y4 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_5140_continuous__on__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [S2: set @ A,F3: A > B,G3: A > C] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ C @ S2 @ G3 )
=> ( topolo81223032696312382ous_on @ A @ ( product_prod @ B @ C ) @ S2
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ).
% continuous_on_Pair
thf(fact_5141_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_5142_continuous__on__mult__const,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [S2: set @ A,C3: A] : ( topolo81223032696312382ous_on @ A @ A @ S2 @ ( times_times @ A @ C3 ) ) ) ).
% continuous_on_mult_const
thf(fact_5143_continuous__on__mult,axiom,
! [A: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S2: set @ D,F3: D > A,G3: D > A] :
( ( topolo81223032696312382ous_on @ D @ A @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ A @ S2 @ G3 )
=> ( topolo81223032696312382ous_on @ D @ A @ S2
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ).
% continuous_on_mult
thf(fact_5144_continuous__on__mult_H,axiom,
! [B: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [A6: set @ D,F3: D > B,G3: D > B] :
( ( topolo81223032696312382ous_on @ D @ B @ A6 @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ B @ A6 @ G3 )
=> ( topolo81223032696312382ous_on @ D @ B @ A6
@ ^ [X4: D] : ( times_times @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ).
% continuous_on_mult'
thf(fact_5145_continuous__on__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S2: set @ B,F3: B > A,C3: A] :
( ( topolo81223032696312382ous_on @ B @ A @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ B @ A @ S2
@ ^ [X4: B] : ( times_times @ A @ C3 @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_mult_left
thf(fact_5146_continuous__on__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S2: set @ B,F3: B > A,C3: A] :
( ( topolo81223032696312382ous_on @ B @ A @ S2 @ F3 )
=> ( topolo81223032696312382ous_on @ B @ A @ S2
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C3 ) ) ) ) ).
% continuous_on_mult_right
thf(fact_5147_IVT_H,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 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ F3 )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F3 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT'
thf(fact_5148_IVT2_H,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 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ F3 )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F3 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT2'
thf(fact_5149_continuous__on__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S2: set @ A,F3: A > B,G3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S2 @ G3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( G3 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X4: A] : ( divide_divide @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ) ).
% continuous_on_divide
thf(fact_5150_continuous__on__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S2: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( F3 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X4: A] : ( inverse_inverse @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_on_inverse
thf(fact_5151_continuous__on__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S2: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( F3 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X4: A] : ( sgn_sgn @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_on_sgn
thf(fact_5152_Bseq__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,G3: nat > A] :
( ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) )
=> ( ( bfun @ nat @ A @ G3 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X4: nat] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( at_top @ nat ) ) ) ) ) ).
% Bseq_mult
thf(fact_5153_Bseq__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( bfun @ nat @ A
@ ^ [N5: nat] : ( F3 @ ( suc @ N5 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ).
% Bseq_Suc_iff
thf(fact_5154_continuous__onI__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( dense_order @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,A6: set @ A] :
( ( topolo1002775350975398744n_open @ B @ ( image2 @ A @ B @ F3 @ A6 ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ( member @ A @ Y3 @ A6 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ A6 @ F3 ) ) ) ) ).
% continuous_onI_mono
thf(fact_5155_continuous__on__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: set @ A,F3: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S2 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( cos @ A @ ( F3 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S2
@ ^ [X4: A] : ( tan @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_on_tan
thf(fact_5156_continuous__on__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S2: set @ A,F3: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S2 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( sin @ A @ ( F3 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S2
@ ^ [X4: A] : ( cot @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_on_cot
thf(fact_5157_continuous__on__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A6: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A6 @ F3 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A6 )
=> ( ( cosh @ A @ ( F3 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ C @ A @ A6
@ ^ [X4: C] : ( tanh @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_on_tanh
thf(fact_5158_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X4: nat] : ( times_times @ A @ C3 @ ( F3 @ X4 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_5159_DERIV__atLeastAtMost__imp__continuous__on,axiom,
! [A: $tType] :
( ( ( ord @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,B2: A,F3: A > A] :
( ! [X3: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B2 )
=> ? [Y6: A] : ( has_field_derivative @ A @ F3 @ Y6 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ F3 ) ) ) ).
% DERIV_atLeastAtMost_imp_continuous_on
thf(fact_5160_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [N6: nat] :
! [N5: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N5 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_5161_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [N6: nat] :
! [N5: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N5 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_5162_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 )
=> ( ? [N8: nat] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N3 @ N8 )
=> ! [M: nat] :
( ( ord_less_eq @ nat @ M @ N8 )
=> ( ( ord_less @ nat @ M @ N3 )
=> ( ord_less @ ( set @ A ) @ ( F3 @ M ) @ ( F3 @ N3 ) ) ) ) )
& ! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ( F3 @ N8 )
= ( F3 @ N3 ) ) ) )
=> ( ( F3 @ ( finite_card @ A @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_5163_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [No3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ No3 )
& ! [N5: nat] :
( ( ord_less_eq @ nat @ No3 @ N5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N5 ) @ L5 ) @ R ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_5164_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( inj_on @ real @ real
@ ^ [Y4: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y4 ) @ ( power_power @ real @ ( abs_abs @ real @ Y4 ) @ N ) )
@ ( top_top @ ( set @ real ) ) ) ) ).
% inj_sgn_power
thf(fact_5165_Sup__lessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_lessThan @ A @ Y ) )
= Y ) ) ).
% Sup_lessThan
thf(fact_5166_Sup__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_atMost @ A @ Y ) )
= Y ) ) ).
% Sup_atMost
thf(fact_5167_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastAtMost
thf(fact_5168_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastAtMost
thf(fact_5169_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastLessThan
thf(fact_5170_dist__0__norm,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( real_V557655796197034286t_dist @ A @ ( zero_zero @ A ) @ X )
= ( real_V7770717601297561774m_norm @ A @ X ) ) ) ).
% dist_0_norm
thf(fact_5171_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_5172_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_5173_Sup__atLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_atLeast @ A @ X ) )
= ( top_top @ A ) ) ) ).
% Sup_atLeast
thf(fact_5174_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_5175_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X ) )
= ( top_top @ A ) ) ) ) ).
% Sup_greaterThanAtLeast
thf(fact_5176_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A
@ ^ [B5: A] : ( divide_divide @ A @ B5 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_5177_dist__scaleR,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: real,A3: A,Y: real] :
( ( real_V557655796197034286t_dist @ A @ ( real_V8093663219630862766scaleR @ A @ X @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ Y @ A3 ) )
= ( times_times @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y ) ) @ ( real_V7770717601297561774m_norm @ A @ A3 ) ) ) ) ).
% dist_scaleR
thf(fact_5178_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X8: set @ A,A3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ A3 ) )
=> ( ! [Y3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ X5 @ Y3 ) )
=> ( ord_less_eq @ A @ A3 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A3 ) ) ) ) ).
% cSup_eq
thf(fact_5179_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z3: A,X8: set @ A] :
( ( member @ A @ Z3 @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= Z3 ) ) ) ) ).
% cSup_eq_maximum
thf(fact_5180_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,A6: set @ A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ A6 ) ) ) ).
% inj_on_mult
thf(fact_5181_subset__image__inj,axiom,
! [A: $tType,B: $tType,S3: set @ A,F3: B > A,T6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( image2 @ B @ A @ F3 @ T6 ) )
= ( ? [U3: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U3 @ T6 )
& ( inj_on @ B @ A @ F3 @ U3 )
& ( S3
= ( image2 @ B @ A @ F3 @ U3 ) ) ) ) ) ).
% subset_image_inj
thf(fact_5182_norm__conv__dist,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A )
= ( ^ [X4: A] : ( real_V557655796197034286t_dist @ A @ X4 @ ( zero_zero @ A ) ) ) ) ) ).
% norm_conv_dist
thf(fact_5183_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X8 ) @ Z3 ) ) ) ) ).
% cSup_least
thf(fact_5184_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ A3 ) )
=> ( ! [Y3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ X5 @ Y3 ) )
=> ( ord_less_eq @ A @ A3 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A3 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_5185_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_5186_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_5187_inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B6: set @ A] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ B6 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ A @ A @ ( inf_inf @ A @ A3 ) @ B6 ) ) ) ) ).
% inf_Sup
thf(fact_5188_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B6: set @ A,A3: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B6 ) @ A3 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B6 )
=> ( ( inf_inf @ A @ X4 @ A3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_5189_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 ) )
=> ~ ! [F9: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F9 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F9 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ F9 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_5190_card__Union__le__sum__card,axiom,
! [A: $tType,U4: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U4 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U4 ) ) ).
% card_Union_le_sum_card
thf(fact_5191_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,B6: set @ B,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ B6 ) ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( inf_inf @ A @ ( F3 @ B5 ) @ A3 )
@ B6 ) ) ) ) ).
% SUP_inf
thf(fact_5192_Sup__inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B6: set @ A,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B6 ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image2 @ A @ A
@ ^ [B5: A] : ( inf_inf @ A @ B5 @ A3 )
@ B6 ) ) ) ) ).
% Sup_inf
thf(fact_5193_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F3: B > A,B6: set @ B] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ B6 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( inf_inf @ A @ A3 @ ( F3 @ B5 ) )
@ B6 ) ) ) ) ).
% inf_SUP
thf(fact_5194_SUP__inf__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,A6: set @ B,G3: C > A,B6: set @ C] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A7: B] :
( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [B5: C] : ( inf_inf @ A @ ( F3 @ A7 ) @ ( G3 @ B5 ) )
@ B6 ) )
@ A6 ) ) ) ) ).
% SUP_inf_distrib2
thf(fact_5195_UN__lessThan__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_lessThan_UNIV
thf(fact_5196_UN__atMost__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atMost_UNIV
thf(fact_5197_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_5198_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F3: B > A,M5: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ M5 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ M5 ) ) ) ) ).
% cSUP_least
thf(fact_5199_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S3 ) ) @ A3 ) ) ) ) ).
% cSup_abs_le
thf(fact_5200_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U4: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ U4 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U4 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U4 ) ) ) ).
% card_Union_le_sum_card_weak
thf(fact_5201_UN__atLeast__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atLeast @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atLeast_UNIV
thf(fact_5202_UN__UN__finite__eq,axiom,
! [A: $tType,A6: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [N5: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_5203_UN__le__add__shift__strict,axiom,
! [A: $tType,M5: nat > ( set @ A ),K2: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I: nat] : ( M5 @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M5 @ ( set_or7035219750837199246ssThan @ nat @ K2 @ ( plus_plus @ nat @ N @ K2 ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_5204_UN__le__add__shift,axiom,
! [A: $tType,M5: nat > ( set @ A ),K2: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I: nat] : ( M5 @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M5 @ ( set_or1337092689740270186AtMost @ nat @ K2 @ ( plus_plus @ nat @ N @ K2 ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_5205_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S3 ) @ L ) ) @ E3 ) ) ) ) ).
% cSup_asclose
thf(fact_5206_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M10: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M10 @ M )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M10 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M ) @ ( X8 @ N3 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% metric_CauchyI
thf(fact_5207_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,E3: real] :
( ( topolo3814608138187158403Cauchy @ A @ X8 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M8: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M8 @ M3 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ M8 @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M3 ) @ ( X8 @ N4 ) ) @ E3 ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_5208_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S6: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N6: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ N6 @ N5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S6 @ N5 ) @ ( S6 @ N6 ) ) @ E4 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_5209_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X7: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ M9 @ M6 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ M9 @ N5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ M6 ) @ ( X7 @ N5 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_5210_Sup__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: nat > A,S2: set @ A,A3: A] :
( ! [N3: nat] : ( member @ A @ ( B2 @ N3 ) @ S2 )
=> ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ A3 @ ( complete_Sup_Sup @ A @ S2 ) ) ) ) ) ).
% Sup_lim
thf(fact_5211_UN__finite__subset,axiom,
! [A: $tType,A6: nat > ( set @ A ),C5: set @ A] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ).
% UN_finite_subset
thf(fact_5212_UN__finite2__eq,axiom,
! [A: $tType,A6: nat > ( set @ A ),B6: nat > ( set @ A ),K2: nat] :
( ! [N3: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K2 ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_5213_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
@ ^ [I: A] : ( finite_card @ B @ ( A6 @ I ) )
@ I5 ) ) ) ).
% card_UN_le
thf(fact_5214_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A6 ) @ B6 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B6 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_5215_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ A,B6: set @ B] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ B6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B6 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_5216_card__le__inj,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B6 ) )
=> ? [F2: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A6 ) @ B6 )
& ( inj_on @ A @ B @ F2 @ A6 ) ) ) ) ) ).
% card_le_inj
thf(fact_5217_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] :
! [M6: nat] :
( ( ord_less_eq @ nat @ M9 @ M6 )
=> ! [N5: nat] :
( ( ord_less @ nat @ M6 @ N5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F4 @ M6 ) @ ( F4 @ N5 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_5218_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M10: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M10 @ M )
=> ! [N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M ) @ ( X8 @ N3 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% CauchyI'
thf(fact_5219_card__partition,axiom,
! [A: $tType,C5: set @ ( set @ A ),K2: nat] :
( ( finite_finite2 @ ( set @ A ) @ C5 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) )
=> ( ! [C2: set @ A] :
( ( member @ ( set @ A ) @ C2 @ C5 )
=> ( ( finite_card @ A @ C2 )
= K2 ) )
=> ( ! [C1: set @ A,C22: set @ A] :
( ( member @ ( set @ A ) @ C1 @ C5 )
=> ( ( member @ ( set @ A ) @ C22 @ C5 )
=> ( ( C1 != C22 )
=> ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K2 @ ( finite_card @ ( set @ A ) @ C5 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_5220_UN__finite2__subset,axiom,
! [A: $tType,A6: nat > ( set @ A ),B6: nat > ( set @ A ),K2: nat] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K2 ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_5221_UN__le__eq__Un0,axiom,
! [A: $tType,M5: nat > ( set @ A ),N: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M5 @ ( set_ord_atMost @ nat @ N ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M5 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M5 @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_5222_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A,R3: real] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ No @ N4 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N4 ) @ L5 ) @ R3 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_5223_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A] :
( ! [R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
=> ? [No2: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No2 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N3 ) @ L5 ) @ R4 ) ) )
=> ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_5224_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [No3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ No3 @ N5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N5 ) @ L5 ) @ R ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_5225_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_5226_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X7: nat > A] :
! [J: nat] :
? [M9: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ M9 @ M6 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ M9 @ N5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ M6 ) @ ( X7 @ N5 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_5227_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,T2: list @ ( product_prod @ A @ C ),K2: A,X: C] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T2 @ K2 )
= ( some @ C @ X ) )
=> ( ( map_of @ B @ C
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
@ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
@ ^ [K3: A] : ( product_Pair @ B @ C @ ( F3 @ K3 ) ) )
@ T2 )
@ ( F3 @ K2 ) )
= ( some @ C @ X ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_5228_SUP__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: A,B6: A] :
( ( sup_sup @ A @ A6
@ ( complete_Sup_Sup @ A
@ ( image2 @ nat @ A
@ ^ [X4: nat] : B6
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( sup_sup @ A @ A6 @ B6 ) ) ) ).
% SUP_nat_binary
thf(fact_5229_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_5230_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_5231_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_5232_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 )
@ ^ [I: set @ ( product_prod @ A @ B ),X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ I )
@ S3 ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_5233_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R3: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I: C,X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( R3 @ I ) )
@ S3 ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_5234_inj__list__encode,axiom,
! [A6: set @ ( list @ nat )] : ( inj_on @ ( list @ nat ) @ nat @ nat_list_encode @ A6 ) ).
% inj_list_encode
thf(fact_5235_inj__list__decode,axiom,
! [A6: set @ nat] : ( inj_on @ nat @ ( list @ nat ) @ nat_list_decode @ A6 ) ).
% inj_list_decode
thf(fact_5236_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_5237_inj__on__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N7: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N7 ) ) ).
% inj_on_of_nat
thf(fact_5238_inj__Suc,axiom,
! [N7: set @ nat] : ( inj_on @ nat @ nat @ suc @ N7 ) ).
% inj_Suc
thf(fact_5239_inj__Some,axiom,
! [A: $tType,A6: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A6 ) ).
% inj_Some
thf(fact_5240_inj__prod__decode,axiom,
! [A6: set @ nat] : ( inj_on @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ A6 ) ).
% inj_prod_decode
thf(fact_5241_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F3: A > B,X8: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ ( F3 @ X4 ) )
@ X8 ) ).
% inj_on_convol_ident
thf(fact_5242_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_5243_inj__on__diff__nat,axiom,
! [N7: set @ nat,K2: nat] :
( ! [N3: nat] :
( ( member @ nat @ N3 @ N7 )
=> ( ord_less_eq @ nat @ K2 @ N3 ) )
=> ( inj_on @ nat @ nat
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ K2 )
@ N7 ) ) ).
% inj_on_diff_nat
thf(fact_5244_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S7: set @ ( A > B > $o ),X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S7 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_5245_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 )
@ ^ [I: A,J: B] : ( product_Pair @ B @ A @ J @ I ) )
@ A6 ) ).
% swap_inj_on
thf(fact_5246_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_5247_inj__on__nth,axiom,
! [A: $tType,Xs2: list @ A,I5: set @ nat] :
( ( distinct @ A @ Xs2 )
=> ( ! [X3: nat] :
( ( member @ nat @ X3 @ I5 )
=> ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs2 ) @ I5 ) ) ) ).
% inj_on_nth
thf(fact_5248_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_5249_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_5250_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F3: A > A,S2: A] :
( ( ( compow @ ( A > A ) @ N @ F3 @ S2 )
= S2 )
=> ( ! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ( ( compow @ ( A > A ) @ M @ F3 @ S2 )
!= S2 ) ) )
=> ( inj_on @ nat @ A
@ ^ [K3: nat] : ( compow @ ( A > A ) @ K3 @ F3 @ S2 )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% inj_on_funpow_least
thf(fact_5251_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,X: A] :
( ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [Y3: A] :
( ! [Z5: A] :
( ( member @ A @ Z5 @ A6 )
=> ( ord_less_eq @ A @ Z5 @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ A6 )
= X ) ) ) ) ).
% Sup_eqI
thf(fact_5252_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ B6 )
& ( ord_less_eq @ A @ A5 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ).
% Sup_mono
thf(fact_5253_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,Z3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ Z3 ) ) ) ).
% Sup_least
thf(fact_5254_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A6: set @ A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ).
% Sup_upper
thf(fact_5255_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_5256_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A6: set @ A,V3: A] :
( ( member @ A @ U @ A6 )
=> ( ( ord_less_eq @ A @ V3 @ U )
=> ( ord_less_eq @ A @ V3 @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% Sup_upper2
thf(fact_5257_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,A6: set @ A] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A6 ) )
= ( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X )
=> ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ord_less @ A @ Y4 @ X4 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_5258_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B6: set @ C,F3: B > A,G3: C > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B6 )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ X5 ) ) ) )
=> ( ! [J3: C] :
( ( member @ C @ J3 @ B6 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A6 )
& ( ord_less_eq @ A @ ( G3 @ J3 ) @ ( F3 @ X5 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_5259_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,U: A] :
( ! [V2: A] :
( ( member @ A @ V2 @ A6 )
=> ( ord_less_eq @ A @ U @ V2 ) )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_5260_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ).
% Sup_subset_mono
thf(fact_5261_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F3: B > A,X: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ X ) )
=> ( ! [Y3: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ I4 ) @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
= X ) ) ) ) ).
% SUP_eqI
thf(fact_5262_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B6: set @ C,F3: B > A,G3: C > A] :
( ! [N3: B] :
( ( member @ B @ N3 @ A6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B6 )
& ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G3 @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) ) ) ) ).
% SUP_mono
thf(fact_5263_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_5264_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G3: B > A,A6: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) )
=> ( 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_5265_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A6: set @ B,F3: B > A] :
( ( member @ B @ I2 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ).
% SUP_upper
thf(fact_5266_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_5267_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A6: set @ B,U: A,F3: B > A] :
( ( member @ B @ I2 @ A6 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ I2 ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_5268_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,F3: B > A,A6: set @ B] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) )
= ( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X )
=> ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less @ A @ Y4 @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_5269_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_5270_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B6: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_5271_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B6: set @ B,F3: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A6 @ B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_5272_card__UNION,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A6 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A6 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( ( 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_5273_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A6: set @ A,F3: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( member @ A @ Y3 @ A6 )
=> ( ( F3 @ X3 )
!= ( F3 @ Y3 ) ) ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ( member @ A @ Y3 @ A6 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) )
=> ( inj_on @ A @ B @ F3 @ A6 ) ) ) ) ).
% linorder_inj_onI
thf(fact_5274_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_5275_Inf__greaterThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_greaterThan @ A @ X ) )
= X ) ) ).
% Inf_greaterThan
thf(fact_5276_Inf__atLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atLeast @ A @ X ) )
= X ) ) ).
% Inf_atLeast
thf(fact_5277_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_5278_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastAtMost
thf(fact_5279_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastAtMost
thf(fact_5280_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastLessThan
thf(fact_5281_Inf__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% Inf_atMost
thf(fact_5282_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_5283_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_5284_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_5285_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_5286_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z3: A,X8: set @ A] :
( ( member @ A @ Z3 @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= Z3 ) ) ) ) ).
% cInf_eq_minimum
thf(fact_5287_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X8: set @ A,A3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ A3 @ X3 ) )
=> ( ! [Y3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ Y3 @ X5 ) )
=> ( ord_less_eq @ A @ Y3 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A3 ) ) ) ) ).
% cInf_eq
thf(fact_5288_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,X: A] :
( ! [I3: A] :
( ( member @ A @ I3 @ A6 )
=> ( ord_less_eq @ A @ X @ I3 ) )
=> ( ! [Y3: A] :
( ! [I4: A] :
( ( member @ A @ I4 @ A6 )
=> ( ord_less_eq @ A @ Y3 @ I4 ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ A6 )
= X ) ) ) ) ).
% Inf_eqI
thf(fact_5289_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ A,A6: set @ A] :
( ! [B4: A] :
( ( member @ A @ B4 @ B6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ( ord_less_eq @ A @ X5 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ).
% Inf_mono
thf(fact_5290_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A6: set @ A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ X ) ) ) ).
% Inf_lower
thf(fact_5291_Inf__lower2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A6: set @ A,V3: A] :
( ( member @ A @ U @ A6 )
=> ( ( ord_less_eq @ A @ U @ V3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ V3 ) ) ) ) ).
% Inf_lower2
thf(fact_5292_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_5293_Inf__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,Z3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ).
% Inf_greatest
thf(fact_5294_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A6: set @ A,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ X )
= ( ! [Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ord_less @ A @ X4 @ Y4 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_5295_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B6: set @ C,G3: C > A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B6 )
& ( ord_less_eq @ A @ ( G3 @ X5 ) @ ( F3 @ I3 ) ) ) )
=> ( ! [J3: C] :
( ( member @ C @ J3 @ B6 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A6 )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ J3 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) ) ) ) ) ).
% INF_eq
thf(fact_5296_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ).
% cInf_greatest
thf(fact_5297_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ A3 @ X3 ) )
=> ( ! [Y3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X8 )
=> ( ord_less_eq @ A @ Y3 @ X5 ) )
=> ( ord_less_eq @ A @ Y3 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A3 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_5298_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,U: A] :
( ! [V2: A] :
( ( member @ A @ V2 @ A6 )
=> ( ord_less_eq @ A @ V2 @ U ) )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_5299_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X ) ) ) ) ).
% cInf_le_finite
thf(fact_5300_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ A,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ).
% Inf_superset_mono
thf(fact_5301_sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B6: set @ A] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ B6 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ A @ A @ ( sup_sup @ A @ A3 ) @ B6 ) ) ) ) ).
% sup_Inf
thf(fact_5302_Inf__sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B6: set @ A,A3: A] :
( ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B6 ) @ A3 )
= ( top_top @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B6 )
=> ( ( sup_sup @ A @ X4 @ A3 )
= ( top_top @ A ) ) ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_5303_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,X: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less_eq @ A @ X @ ( F3 @ I3 ) ) )
=> ( ! [Y3: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ( ord_less_eq @ A @ Y3 @ ( F3 @ I4 ) ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
= X ) ) ) ) ).
% INF_eqI
thf(fact_5304_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ B,A6: set @ C,F3: C > A,G3: B > A] :
( ! [M: B] :
( ( member @ B @ M @ B6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ A6 )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ M ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ).
% INF_mono
thf(fact_5305_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A6: set @ B,F3: B > A] :
( ( member @ B @ I2 @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( F3 @ I2 ) ) ) ) ).
% INF_lower
thf(fact_5306_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G3: B > A,A6: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) )
=> ( 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_5307_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A6: set @ B,F3: B > A,U: A] :
( ( member @ B @ I2 @ A6 )
=> ( ( ord_less_eq @ A @ ( F3 @ I2 ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_5308_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_5309_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_5310_INF__sup__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,A6: set @ B,G3: C > A,B6: set @ C] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A7: B] :
( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [B5: C] : ( sup_sup @ A @ ( F3 @ A7 ) @ ( G3 @ B5 ) )
@ B6 ) )
@ A6 ) ) ) ) ).
% INF_sup_distrib2
thf(fact_5311_sup__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F3: B > A,B6: set @ B] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ B6 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( sup_sup @ A @ A3 @ ( F3 @ B5 ) )
@ B6 ) ) ) ) ).
% sup_INF
thf(fact_5312_Inf__sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B6: set @ A,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B6 ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [B5: A] : ( sup_sup @ A @ B5 @ A3 )
@ B6 ) ) ) ) ).
% Inf_sup
thf(fact_5313_INF__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,B6: set @ B,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ B6 ) ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( sup_sup @ A @ ( F3 @ B5 ) @ A3 )
@ B6 ) ) ) ) ).
% INF_sup
thf(fact_5314_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_5315_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A6: set @ B,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ X )
= ( ! [Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less @ A @ ( F3 @ X4 ) @ Y4 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_5316_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,M2: A,F3: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ M2 @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ M2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_5317_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_5318_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_5319_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S3 ) ) @ A3 ) ) ) ) ).
% cInf_abs_ge
thf(fact_5320_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B6: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B6 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_5321_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ B,A6: set @ B,F3: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B6 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_5322_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A8: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B5: A] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ord_less_eq @ A @ B5 @ X4 ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_5323_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A8: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B5: A] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ord_less_eq @ A @ X4 @ B5 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_5324_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_5325_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_5326_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_5327_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_5328_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S3 ) @ L ) ) @ E3 ) ) ) ) ).
% cInf_asclose
thf(fact_5329_Inf__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: nat > A,S2: set @ A,A3: A] :
( ! [N3: nat] : ( member @ A @ ( B2 @ N3 ) @ S2 )
=> ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ S2 ) @ A3 ) ) ) ) ).
% Inf_lim
thf(fact_5330_mono__bij__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( comple5582772986160207858norder @ B ) )
=> ! [F3: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( F3 @ ( complete_Inf_Inf @ A @ A6 ) )
= ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ A6 ) ) ) ) ) ) ).
% mono_bij_Inf
thf(fact_5331_INF__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: A,B6: A] :
( ( inf_inf @ A @ A6
@ ( complete_Inf_Inf @ A
@ ( image2 @ nat @ A
@ ^ [X4: nat] : B6
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( inf_inf @ A @ A6 @ B6 ) ) ) ).
% INF_nat_binary
thf(fact_5332_INF__SUP,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P2: C > B > A] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y4: B] :
( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [X4: C] : ( P2 @ X4 @ Y4 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( B > C ) @ A
@ ^ [F4: B > C] :
( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( P2 @ ( F4 @ X4 ) @ X4 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% INF_SUP
thf(fact_5333_SUP__INF,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P2: C > B > A] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y4: B] :
( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [X4: C] : ( P2 @ X4 @ Y4 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( B > C ) @ A
@ ^ [X4: B > C] :
( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y4: B] : ( P2 @ ( X4 @ Y4 ) @ Y4 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% SUP_INF
thf(fact_5334_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_5335_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_5336_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_5337_Gcd__eq__Max,axiom,
! [M5: set @ nat] :
( ( finite_finite2 @ nat @ M5 )
=> ( ( M5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( gcd_Gcd @ nat @ M5 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image2 @ nat @ ( set @ nat )
@ ^ [M6: nat] :
( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ M6 ) )
@ M5 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_5338_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F3: C > A,X: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_Pair @ C @ B @ X @ Y ) )
= ( product_Pair @ A @ B @ ( F3 @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_5339_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B,X: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_Pair @ A @ C @ X @ Y ) )
= ( product_Pair @ A @ B @ X @ ( F3 @ Y ) ) ) ).
% apsnd_conv
thf(fact_5340_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > A,X: product_prod @ C @ B,G3: C > A] :
( ( ( product_apfst @ C @ A @ B @ F3 @ X )
= ( product_apfst @ C @ A @ B @ G3 @ X ) )
= ( ( F3 @ ( product_fst @ C @ B @ X ) )
= ( G3 @ ( product_fst @ C @ B @ X ) ) ) ) ).
% apfst_eq_conv
thf(fact_5341_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > A,X: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F3 @ X ) )
= ( F3 @ ( product_fst @ C @ B @ X ) ) ) ).
% fst_apfst
thf(fact_5342_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F3: C > B,X: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F3 @ X ) )
= ( product_snd @ C @ A @ X ) ) ).
% snd_apfst
thf(fact_5343_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F3 @ X ) )
= ( product_fst @ A @ C @ X ) ) ).
% fst_apsnd
thf(fact_5344_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C,G3: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F3 @ X )
= ( product_apsnd @ C @ B @ A @ G3 @ X ) )
= ( ( F3 @ ( product_snd @ A @ C @ X ) )
= ( G3 @ ( product_snd @ A @ C @ X ) ) ) ) ).
% apsnd_eq_conv
thf(fact_5345_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,X: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F3 @ X ) )
= ( F3 @ ( product_snd @ B @ C @ X ) ) ) ).
% snd_apsnd
thf(fact_5346_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_5347_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_5348_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F3: C > B,G3: D > A,X: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apfst @ D @ A @ C @ G3 @ X ) )
= ( product_Pair @ A @ B @ ( G3 @ ( product_fst @ D @ C @ X ) ) @ ( F3 @ ( product_snd @ D @ C @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_5349_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F3: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_apsnd @ D @ B @ C @ G3 @ X ) )
= ( product_Pair @ A @ B @ ( F3 @ ( product_fst @ C @ D @ X ) ) @ ( G3 @ ( product_snd @ C @ D @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_5350_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F3: C > B,G3: D > A,P: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apfst @ D @ A @ C @ G3 @ P ) )
= ( product_apfst @ D @ A @ B @ G3 @ ( product_apsnd @ C @ B @ D @ F3 @ P ) ) ) ).
% apsnd_apfst_commute
thf(fact_5351_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_5352_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ B6 )
& ( ord_less_eq @ A @ X3 @ Xa ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B6 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ A6 )
& ( ord_less_eq @ A @ X3 @ Xa ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= ( lattic643756798349783984er_Max @ A @ B6 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_5353_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A6 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( member @ A @ X @ A6 )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= X ) ) ) ) ) ).
% Max_eqI
thf(fact_5354_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ).
% Max_ge
thf(fact_5355_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_5356_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( ord_less_eq @ A @ A5 @ X ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X ) ) ) ) ) ).
% Max.boundedI
thf(fact_5357_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X )
=> ! [A9: A] :
( ( member @ A @ A9 @ A6 )
=> ( ord_less_eq @ A @ A9 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_5358_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_5359_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ord_less_eq @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_5360_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_5361_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 @ ( insert @ A @ A3 @ A6 ) )
= A3 ) ) ) ) ).
% Max_insert2
thf(fact_5362_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 )
@ ^ [I: set @ ( product_prod @ A @ B ),X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ I )
@ S3 ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% INF_Int_eq2
thf(fact_5363_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R3: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I: C,X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( R3 @ I ) )
@ S3 ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_5364_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ ( lattic643756798349783984er_Max @ A @ B6 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_5365_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M5: set @ A,N7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M5 @ N7 )
=> ( ( M5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M5 ) @ ( lattic643756798349783984er_Max @ A @ N7 ) ) ) ) ) ) ).
% Max_mono
thf(fact_5366_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X7: set @ nat] :
( if @ nat
@ ( X7
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X7 ) ) ) ) ).
% Sup_nat_def
thf(fact_5367_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_5368_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M6: nat,N5: nat] :
( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N5 ) @ M6 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_5369_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S7: set @ ( A > B > $o ),X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S7 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_5370_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_5371_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Y
= ( Xa2
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
=> ( Y
= ( ~ ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ 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 ) ) ) @ TreeList @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_5372_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2
!= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ 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 ) ) ) @ TreeList @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_5373_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2
= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ 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 ) ) ) @ TreeList @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_5374_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,Y4: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y4 ) )
& ( X4 != Y4 ) ) ) ) ) ).
% open_diagonal_complement
thf(fact_5375_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_5376_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_5377_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_5378_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_5379_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_5380_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,Y4: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y4 ) )
& ( ord_less @ A @ X4 @ Y4 ) ) ) ) ) ).
% open_subdiagonal
thf(fact_5381_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,Y4: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y4 ) )
& ( ord_less @ A @ Y4 @ X4 ) ) ) ) ) ).
% open_superdiagonal
thf(fact_5382_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R: 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] :
? [Us3: list @ A,Z4: A,Z8: A,Vs3: list @ A] :
( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ Z4 @ Vs3 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ Z8 ) @ R )
& ( Ys3
= ( append @ A @ Us3 @ ( cons @ A @ Z8 @ Vs3 ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_5383_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_5384_case__optionE,axiom,
! [A: $tType,P2: $o,Q: A > $o,X: option @ A] :
( ( case_option @ $o @ A @ P2 @ Q @ X )
=> ( ( ( X
= ( none @ A ) )
=> ~ P2 )
=> ~ ! [Y3: A] :
( ( X
= ( some @ A @ Y3 ) )
=> ~ ( Q @ Y3 ) ) ) ) ).
% case_optionE
thf(fact_5385_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ? [Xys2: list @ A,X4: A,Y4: A,Xs5: list @ A,Ys6: list @ A] :
( ( Xs
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs5 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y4 @ Ys6 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_5386_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I: nat] :
( ( Uu3
= ( nth @ A @ Xs @ I ) )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_5387_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P2: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P2 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P2 @ F1 ) )
| ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
& ~ ( P2 @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_5388_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P2: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P2 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P2 @ F1 ) )
& ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
=> ( P2 @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_5389_set__nths,axiom,
! [A: $tType,Xs2: list @ A,I5: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs2 @ I5 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I: nat] :
( ( Uu3
= ( nth @ A @ Xs2 @ I ) )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( member @ nat @ I @ I5 ) ) ) ) ).
% set_nths
thf(fact_5390_funpow__inj__finite,axiom,
! [A: $tType,P: A > A,X: A] :
( ( inj_on @ A @ A @ P @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y4: A] :
? [N5: nat] :
( Y4
= ( compow @ ( A > A ) @ N5 @ P @ X ) ) ) )
=> ~ ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( compow @ ( A > A ) @ N3 @ P @ X )
!= X ) ) ) ) ).
% funpow_inj_finite
thf(fact_5391_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg3 )
= ( ( Deg = Deg3 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( 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 ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X7 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ 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 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_5392_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
=> ( Xa2
= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ 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 ) ) ) @ TreeList @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_5393_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
=> ( Xa2
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ 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 ) ) ) @ TreeList @ X4 )
=> ( ( ord_less @ nat @ Mi3 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_5394_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Y
= ( Xa2
= ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( Y
= ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ 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 ) ) ) @ TreeList @ 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 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_5395_Sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A6: set @ ( set @ A )] :
( ( complete_Sup_Sup @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) @ A6 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ 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 ) ) ) ) ) ) ) ) ).
% Sup_Inf
thf(fact_5396_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_5397_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_5398_SUP__INF__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G3: B > A,A6: set @ ( set @ B )] :
( ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [X4: set @ B] : ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ X4 ) )
@ A6 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [X4: set @ B] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ X4 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu3: set @ B] :
? [F4: ( set @ B ) > B] :
( ( Uu3
= ( image2 @ ( set @ B ) @ B @ F4 @ A6 ) )
& ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ A6 )
=> ( member @ B @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% SUP_INF_set
thf(fact_5399_INF__SUP__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G3: B > A,A6: set @ ( set @ B )] :
( ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [B8: set @ B] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ B8 ) )
@ A6 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [B8: set @ B] : ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B8 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu3: set @ B] :
? [F4: ( set @ B ) > B] :
( ( Uu3
= ( image2 @ ( set @ B ) @ B @ F4 @ A6 ) )
& ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ A6 )
=> ( member @ B @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% INF_SUP_set
thf(fact_5400_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_5401_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R: set @ ( product_prod @ A @ A ),N5: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N5 )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N5 )
& ? [Xys2: list @ A,X4: A,Y4: A,Xs5: list @ A,Ys6: list @ A] :
( ( Xs
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs5 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y4 @ Ys6 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_5402_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X4: list @ A,Y4: list @ A] :
? [A7: A,V4: list @ A] :
( ( Y4
= ( append @ A @ X4 @ ( cons @ A @ A7 @ V4 ) ) )
| ? [U2: list @ A,B5: A,C4: A,W3: list @ A,Z4: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C4 ) @ R )
& ( X4
= ( append @ A @ U2 @ ( cons @ A @ B5 @ W3 ) ) )
& ( Y4
= ( append @ A @ U2 @ ( cons @ A @ C4 @ Z4 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_5403_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X: list @ A,B2: A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A3 @ X ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R3 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
| ( ( A3 = B2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_5404_lexn_Osimps_I1_J,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( lexn @ A @ R3 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ).
% lexn.simps(1)
thf(fact_5405_lexord__linear,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
( ! [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R3 )
| ( A5 = B4 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R3 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
| ( X = Y )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_linear
thf(fact_5406_lexord__irreflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_irreflexive
thf(fact_5407_lexord__partial__trans,axiom,
! [A: $tType,Xs2: list @ A,R3: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs: list @ A] :
( ! [X3: A,Y3: A,Z: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z ) @ R3 ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lexord @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Zs ) @ ( lexord @ A @ R3 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_5408_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U: list @ A,V3: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V3 ) ) @ ( lexord @ A @ R3 ) )
=> ( ! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V3 ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_append_leftD
thf(fact_5409_lexord__sufE,axiom,
! [A: $tType,Xs2: list @ A,Zs: list @ A,Ys: list @ A,Qs: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Zs ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R3 ) )
=> ( ( Xs2 != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lexord @ A @ R3 ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_5410_lexord__lex,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lex @ A @ R3 ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
& ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_5411_lexn__length,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),N: nat] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( lexn @ A @ R3 @ N ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= N )
& ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ) ).
% lexn_length
thf(fact_5412_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A3 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_append_left_rightI
thf(fact_5413_lexord__same__pref__iff,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A,Zs: list @ A,R3: 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 @ Zs ) ) @ ( lexord @ A @ R3 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R3 ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_5414_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W2: list @ A,R3: set @ ( product_prod @ A @ A ),V3: list @ A,Z3: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W2 ) @ ( lexord @ A @ R3 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W2 ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V3 ) @ ( append @ A @ W2 @ Z3 ) ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_sufI
thf(fact_5415_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,Y4: A] :
( ( ord_less @ nat @ ( F4 @ X4 ) @ ( F4 @ Y4 ) )
| ( ( ord_less_eq @ nat @ ( F4 @ X4 ) @ ( F4 @ Y4 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_5416_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_5417_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q2: product_prod @ A @ B,F3: C > A,P: product_prod @ C @ B] :
( ( Q2
= ( product_apfst @ C @ A @ B @ F3 @ P ) )
=> ~ ! [X3: C,Y3: B] :
( ( P
= ( product_Pair @ C @ B @ X3 @ Y3 ) )
=> ( Q2
!= ( product_Pair @ A @ B @ ( F3 @ X3 ) @ Y3 ) ) ) ) ).
% apfst_convE
thf(fact_5418_mlex__leq,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R2 ) ) ) ) ).
% mlex_leq
thf(fact_5419_mlex__iff,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R2 ) )
= ( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X )
= ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% mlex_iff
thf(fact_5420_mlex__less,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R2 ) ) ) ).
% mlex_less
thf(fact_5421_in__measure,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measure @ A @ F3 ) )
= ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ).
% in_measure
thf(fact_5422_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 )
@ ^ [A7: A,B5: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A14: A,B12: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A14 ) @ Ra )
| ( ( A7 = A14 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B5 @ B12 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_5423_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A8: set @ C,F4: C > A,G4: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A7: C] :
( ( Uu3
= ( product_Pair @ A @ B @ ( F4 @ A7 ) @ ( G4 @ A7 ) ) )
& ( member @ C @ A7 @ A8 ) ) ) ) ) ).
% image2_def
thf(fact_5424_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B,R3: set @ ( product_prod @ A @ A ),S2: 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 @ R3 @ S2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A4 ) @ R3 )
| ( ( A3 = A4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_5425_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F3: B > A,X: B,C3: C,G3: B > C,A6: set @ B] :
( ( B2
= ( F3 @ X ) )
=> ( ( C3
= ( G3 @ X ) )
=> ( ( member @ B @ X @ 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_5426_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,Y8: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y4: B] :
( ( X9 = X4 )
& ( P4 @ X4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y4 ) @ ( R6 @ X4 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_5427_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_5428_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] :
? [I: nat] :
( ( Uu3
= ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Ys @ I ) ) )
& ( ord_less @ nat @ I @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_5429_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_5430_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_5431_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_5432_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% min_top2
thf(fact_5433_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% min_top
thf(fact_5434_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_5435_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_5436_max__min__same_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ X @ ( ord_min @ A @ X @ Y ) )
= X ) ) ).
% max_min_same(1)
thf(fact_5437_max__min__same_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y ) @ X )
= X ) ) ).
% max_min_same(2)
thf(fact_5438_max__min__same_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y ) @ Y )
= Y ) ) ).
% max_min_same(3)
thf(fact_5439_max__min__same_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ( ord_max @ A @ Y @ ( ord_min @ A @ X @ Y ) )
= Y ) ) ).
% max_min_same(4)
thf(fact_5440_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_5441_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_5442_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_5443_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_5444_min__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ A @ V3 ) ) ) ) ) ).
% min_number_of(1)
thf(fact_5445_min__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(3)
thf(fact_5446_min__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(4)
thf(fact_5447_min__0__1_I2_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_min @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(2)
thf(fact_5448_min__0__1_I1_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_min @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(1)
thf(fact_5449_min__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) ) ) ).
% min_0_1(5)
thf(fact_5450_min__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% min_0_1(6)
thf(fact_5451_Int__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atMost @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% Int_atMost
thf(fact_5452_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_5453_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_5454_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 )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ N @ M2 ) ) ) ).
% take_bit_num_simps(3)
thf(fact_5455_min__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) ) ) ) ) ).
% min_number_of(2)
thf(fact_5456_min__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V3 ) )
= ( numeral_numeral @ A @ V3 ) ) ) ) ) ).
% min_number_of(3)
thf(fact_5457_min__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V3: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V3 ) ) ) ) ) ) ).
% min_number_of(4)
thf(fact_5458_same__fstI,axiom,
! [B: $tType,A: $tType,P2: A > $o,X: A,Y9: B,Y: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P2 @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y9 @ Y ) @ ( R2 @ X ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y9 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P2 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_5459_Int__atLeastAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C3 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMost
thf(fact_5460_Int__atLeastAtMostR1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ C3 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMostR1
thf(fact_5461_Int__atLeastAtMostL1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atMost @ A @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMostL1
thf(fact_5462_Int__atLeastLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
= ( set_or7035219750837199246ssThan @ A @ ( ord_max @ A @ A3 @ C3 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastLessThan
thf(fact_5463_Int__greaterThanLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C3 @ D3 ) )
= ( set_or5935395276787703475ssThan @ A @ ( ord_max @ A @ A3 @ C3 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_greaterThanLessThan
thf(fact_5464_Int__greaterThanAtMost,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 ) )
= ( set_or3652927894154168847AtMost @ A @ ( ord_max @ A @ A3 @ C3 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_greaterThanAtMost
thf(fact_5465_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_5466_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_5467_zip__replicate,axiom,
! [A: $tType,B: $tType,I2: nat,X: A,J2: nat,Y: B] :
( ( zip @ A @ B @ ( replicate @ A @ I2 @ X ) @ ( replicate @ B @ J2 @ Y ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I2 @ J2 ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_5468_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_5469_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_5470_greaterThan__Int__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( set_ord_lessThan @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% greaterThan_Int_greaterThan
thf(fact_5471_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A7: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A7 @ B5 ) @ A7 @ B5 ) ) ) ) ).
% min_def_raw
thf(fact_5472_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_5473_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_5474_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_5475_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_5476_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_5477_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B5: A] :
( A7
= ( ord_min @ A @ A7 @ B5 ) ) ) ) ) ).
% min.order_iff
thf(fact_5478_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_5479_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_5480_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B5: A] :
( ( ord_min @ A @ A7 @ B5 )
= A7 ) ) ) ) ).
% min.absorb_iff1
thf(fact_5481_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A7: A] :
( ( ord_min @ A @ A7 @ B5 )
= B5 ) ) ) ) ).
% min.absorb_iff2
thf(fact_5482_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_5483_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_5484_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y ) @ Z3 )
= ( ( ord_less_eq @ A @ X @ Z3 )
| ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% min_le_iff_disj
thf(fact_5485_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_min @ A @ X @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_5486_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_min @ A @ X @ Y )
= X ) ) ) ).
% min_absorb1
thf(fact_5487_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A7: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A7 @ B5 ) @ A7 @ B5 ) ) ) ) ).
% min_def
thf(fact_5488_min__diff,axiom,
! [M2: nat,I2: nat,N: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M2 @ I2 ) @ ( minus_minus @ nat @ N @ I2 ) )
= ( minus_minus @ nat @ ( ord_min @ nat @ M2 @ N ) @ I2 ) ) ).
% min_diff
thf(fact_5489_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X @ Y ) @ Z3 )
= ( ord_min @ A @ ( minus_minus @ A @ X @ Z3 ) @ ( minus_minus @ A @ Y @ Z3 ) ) ) ) ).
% min_diff_distrib_left
thf(fact_5490_nat__mult__min__left,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M2 @ N ) @ Q2 )
= ( ord_min @ nat @ ( times_times @ nat @ M2 @ Q2 ) @ ( times_times @ nat @ N @ Q2 ) ) ) ).
% nat_mult_min_left
thf(fact_5491_nat__mult__min__right,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ M2 @ ( ord_min @ nat @ N @ Q2 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M2 @ N ) @ ( times_times @ nat @ M2 @ Q2 ) ) ) ).
% nat_mult_min_right
thf(fact_5492_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_5493_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X @ Y ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_min
thf(fact_5494_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X @ Y ) @ Z3 )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( plus_plus @ A @ Y @ Z3 ) ) ) ) ).
% min_add_distrib_left
thf(fact_5495_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( plus_plus @ A @ X @ ( ord_min @ A @ Y @ Z3 ) )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ).
% min_add_distrib_right
thf(fact_5496_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_min_eq_max
thf(fact_5497_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_max_eq_min
thf(fact_5498_zip__obtain__same__length,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,P2: ( list @ ( product_prod @ A @ B ) ) > $o] :
( ! [Zs2: list @ A,Ws2: list @ B,N3: nat] :
( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( size_size @ ( list @ B ) @ Ws2 ) )
=> ( ( N3
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( Zs2
= ( take @ A @ N3 @ Xs2 ) )
=> ( ( Ws2
= ( take @ B @ N3 @ Ys ) )
=> ( P2 @ ( zip @ A @ B @ Zs2 @ Ws2 ) ) ) ) ) )
=> ( P2 @ ( zip @ A @ B @ Xs2 @ Ys ) ) ) ).
% zip_obtain_same_length
thf(fact_5499_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ P @ ( ord_max @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P @ X ) @ ( times_times @ A @ P @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ P @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P @ X ) @ ( times_times @ A @ P @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_5500_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ P @ ( ord_min @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P @ X ) @ ( times_times @ A @ P @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ P @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P @ X ) @ ( times_times @ A @ P @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_5501_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P )
= ( ord_max @ A @ ( times_times @ A @ X @ P ) @ ( times_times @ A @ Y @ P ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P )
= ( ord_min @ A @ ( times_times @ A @ X @ P ) @ ( times_times @ A @ Y @ P ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_5502_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P )
= ( ord_min @ A @ ( times_times @ A @ X @ P ) @ ( times_times @ A @ Y @ P ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P )
= ( ord_max @ A @ ( times_times @ A @ X @ P ) @ ( times_times @ A @ Y @ P ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_5503_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P ) @ ( divide_divide @ A @ Y @ P ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P ) @ ( divide_divide @ A @ Y @ P ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_5504_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P ) @ ( divide_divide @ A @ Y @ P ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P ) @ ( divide_divide @ A @ Y @ P ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_5505_min__Suc1,axiom,
! [N: nat,M2: nat] :
( ( ord_min @ nat @ ( suc @ N ) @ M2 )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M7: nat] : ( suc @ ( ord_min @ nat @ N @ M7 ) )
@ M2 ) ) ).
% min_Suc1
thf(fact_5506_min__Suc2,axiom,
! [M2: nat,N: nat] :
( ( ord_min @ nat @ M2 @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M7: nat] : ( suc @ ( ord_min @ nat @ M7 @ N ) )
@ M2 ) ) ).
% min_Suc2
thf(fact_5507_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_5508_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_5509_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_5510_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N5: nat,M6: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N5 @ ( numeral_numeral @ nat @ M6 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N5 @ ( numeral_numeral @ nat @ M6 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_5511_lexord__take__index__conv,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y )
= X ) )
| ? [I: nat] :
( ( ord_less @ nat @ I @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I @ X )
= ( take @ A @ I @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I ) @ ( nth @ A @ Y @ I ) ) @ R3 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_5512_set__relcomp,axiom,
! [B: $tType,C: $tType,A: $tType,Xys: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Xy2: 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 @ Xy2 )
= ( product_fst @ C @ B @ Yz ) )
@ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy2 ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
@ ( nil @ ( product_prod @ A @ B ) ) )
@ Yzs ) )
@ Xys ) ) ) ) ).
% set_relcomp
thf(fact_5513_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_5514_has__derivative__power__int_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,N: int,S3: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A
@ ^ [X4: A] : ( power_int @ A @ X4 @ N )
@ ^ [Y4: A] : ( times_times @ A @ Y4 @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ X @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_power_int'
thf(fact_5515_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,W2: num,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X @ ( numeral_numeral @ A @ W2 ) ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_5516_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W2: num,Y: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_5517_power__int__0__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( M2
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( zero_zero @ A ) ) ) ) ).
% power_int_0_left
thf(fact_5518_power__int__eq__0__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( ( power_int @ A @ X @ N )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( N
!= ( zero_zero @ int ) ) ) ) ) ).
% power_int_eq_0_iff
thf(fact_5519_power__int__mult__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N: num] :
( ( power_int @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% power_int_mult_numeral
thf(fact_5520_power__int__minus__one__mult__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) )
= ( one_one @ A ) ) ) ).
% power_int_minus_one_mult_self
thf(fact_5521_power__int__minus__one__mult__self_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int,B2: A] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ B2 ) )
= B2 ) ) ).
% power_int_minus_one_mult_self'
thf(fact_5522_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N: num] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_5523_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N: num,B2: A] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) @ B2 ) ) ) ).
% power_int_add_numeral2
thf(fact_5524_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_5525_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A,C3: B,R3: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ C3 ) @ ( relcomp @ A @ C @ B @ R3 @ S2 ) )
=> ~ ! [B4: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ B4 ) @ R3 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B4 @ C3 ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_5526_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R3: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R3 @ S2 ) )
=> ~ ! [X3: A,Y3: C,Z: B] :
( ( Xz
= ( product_Pair @ A @ B @ X3 @ Z ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_5527_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B ),C3: C,S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C3 ) @ S2 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ C3 ) @ ( relcomp @ A @ B @ C @ R3 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_5528_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A12: A,A23: C,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A12 @ A23 ) @ ( relcomp @ A @ B @ C @ R3 @ S2 ) )
= ( ? [A7: A,B5: B,C4: C] :
( ( A12 = A7 )
& ( A23 = C4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R3 )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ C4 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_5529_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A12: A,A23: C,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A12 @ A23 ) @ ( relcomp @ A @ B @ C @ R3 @ S2 ) )
=> ~ ! [B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A12 @ B4 ) @ R3 )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ A23 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_5530_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X @ Y ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_mult_distrib
thf(fact_5531_power__int__commutes,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( times_times @ A @ ( power_int @ A @ X @ N ) @ X )
= ( times_times @ A @ X @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_commutes
thf(fact_5532_power__int__mult,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int,N: int] :
( ( power_int @ A @ X @ ( times_times @ int @ M2 @ N ) )
= ( power_int @ A @ ( power_int @ A @ X @ M2 ) @ N ) ) ) ).
% power_int_mult
thf(fact_5533_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_5534_zero__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% zero_less_power_int
thf(fact_5535_power__int__not__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_int_not_zero
thf(fact_5536_zero__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% zero_le_power_int
thf(fact_5537_continuous__on__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S2: set @ A,F3: A > B,N: int] :
( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( F3 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S2
@ ^ [X4: A] : ( power_int @ B @ ( F3 @ X4 ) @ N ) ) ) ) ) ).
% continuous_on_power_int
thf(fact_5538_power__int__0__left__If,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( ( M2
= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( one_one @ A ) ) )
& ( ( M2
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( zero_zero @ A ) ) ) ) ) ).
% power_int_0_left_If
thf(fact_5539_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N7: int,A3: A] :
( ( ord_less_eq @ int @ N @ N7 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N7 ) ) ) ) ) ).
% power_int_increasing
thf(fact_5540_power__int__diff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,M2: int,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2 != N ) )
=> ( ( power_int @ A @ X @ ( minus_minus @ int @ M2 @ N ) )
= ( divide_divide @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% power_int_diff
thf(fact_5541_tendsto__power__int,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A3: A,F5: filter @ B,N: int] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( power_int @ A @ ( F3 @ X4 ) @ N )
@ ( topolo7230453075368039082e_nhds @ A @ ( power_int @ A @ A3 @ N ) )
@ F5 ) ) ) ) ).
% tendsto_power_int
thf(fact_5542_continuous__at__within__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A3: A,S2: set @ A,F3: A > B,N: int] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 ) @ F3 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S2 )
@ ^ [X4: A] : ( power_int @ B @ ( F3 @ X4 ) @ N ) ) ) ) ) ).
% continuous_at_within_power_int
thf(fact_5543_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R: set @ ( product_prod @ A @ C ),S6: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Z4: B] :
? [Y4: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Y4 ) @ R )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y4 @ Z4 ) @ S6 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_5544_differentiable__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S2: set @ A,N: int] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( power_int @ B @ ( F3 @ X4 ) @ N )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% differentiable_power_int
thf(fact_5545_ran__def,axiom,
! [B: $tType,A: $tType] :
( ( ran @ A @ B )
= ( ^ [M6: A > ( option @ B )] :
( collect @ B
@ ^ [B5: B] :
? [A7: A] :
( ( M6 @ A7 )
= ( some @ B @ B5 ) ) ) ) ) ).
% ran_def
thf(fact_5546_continuous__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F5: filter @ A,F3: A > B,N: int] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X4: A] : X4 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X4: A] : ( power_int @ B @ ( F3 @ X4 ) @ N ) ) ) ) ) ).
% continuous_power_int
thf(fact_5547_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N7: int,A3: A] :
( ( ord_less @ int @ N @ N7 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N7 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_5548_power__int__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,N: int] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( power_int @ A @ Y @ N ) ) ) ) ) ) ).
% power_int_mono
thf(fact_5549_power__int__strict__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N: int] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_strict_antimono
thf(fact_5550_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% one_le_power_int
thf(fact_5551_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M2 @ N )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ N ) )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% power_int_add
thf(fact_5552_power__int__minus__left__distrib,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( division_ring @ A )
& ( one @ B )
& ( uminus @ B ) )
=> ! [X: C,A3: A,N: int] :
( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% power_int_minus_left_distrib
thf(fact_5553_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_5554_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_5555_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_5556_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N7: int,A3: A] :
( ( ord_less_eq @ int @ N @ N7 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( A3
!= ( zero_zero @ A ) )
| ( N7
!= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N7 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_5557_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M2: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ M2 @ N ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_5558_power__int__minus__mult,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N
!= ( zero_zero @ int ) ) )
=> ( ( times_times @ A @ ( power_int @ A @ X @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) @ X )
= ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_minus_mult
thf(fact_5559_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ X ) ) ) ) ).
% power_int_add_1
thf(fact_5560_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ X @ ( power_int @ A @ X @ M2 ) ) ) ) ) ).
% power_int_add_1'
thf(fact_5561_DERIV__power__int,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S2: set @ A,N: int] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( power_int @ A @ ( F3 @ X4 ) @ N )
@ ( times_times @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ ( F3 @ X ) @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) @ D3 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_power_int
thf(fact_5562_has__derivative__power__int,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,X: C,F6: C > A,S3: set @ C,N: int] :
( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F3 @ F6 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( power_int @ A @ ( F3 @ X4 ) @ N )
@ ^ [H2: C] : ( times_times @ A @ ( F6 @ H2 ) @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ ( F3 @ X ) @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ).
% has_derivative_power_int
thf(fact_5563_zip__Cons1,axiom,
! [A: $tType,B: $tType,X: A,Xs2: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs2 ) @ Ys )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y4: B,Ys3: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( zip @ A @ B @ Xs2 @ Ys3 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_5564_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_5565_pred__nat__def,axiom,
( pred_nat
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [M6: nat,N5: nat] :
( N5
= ( suc @ M6 ) ) ) ) ) ).
% pred_nat_def
thf(fact_5566_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X3 @ 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_5567_fold__union__pair,axiom,
! [B: $tType,A: $tType,B6: set @ A,X: B,A6: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B6 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y4: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B6 ) )
@ A6 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y4: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y4 ) )
@ A6
@ B6 ) ) ) ).
% fold_union_pair
thf(fact_5568_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( ord_min @ A @ X4 ) @ Y4 ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_5569_length__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs2: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs2 ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ).
% length_insort
thf(fact_5570_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Y: B,Ys: list @ B] :
( ( ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ X @ ( cons @ B @ Y @ Ys ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ Y @ ( linorder_insort_key @ B @ A @ F3 @ X @ Ys ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_5571_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs2 ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 ) ) ) ).
% sorted_insort
thf(fact_5572_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ B,F3: B > A,A3: B] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs2 ) )
=> ( ord_less_eq @ A @ ( F3 @ A3 ) @ ( F3 @ X3 ) ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A3 @ Xs2 )
= ( cons @ B @ A3 @ Xs2 ) ) ) ) ).
% insort_is_Cons
thf(fact_5573_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs2: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs2 ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs2 ) ) ) ) ).
% sorted_insort_key
thf(fact_5574_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R2 )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S3 )
=> ( ( relcomp @ A @ B @ C @ R2 @ 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,Y4: B,A8: 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,A15: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y4 = W3 ) @ ( insert @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Z4 ) @ A15 ) @ A15 ) )
@ A8
@ S3 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R2 ) ) ) ) ).
% relcomp_fold
thf(fact_5575_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,R2: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X @ R2 ) @ 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,A15: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z4 ) @ A15 )
@ A15 ) )
@ ( relcomp @ C @ A @ B @ R2 @ S3 )
@ S3 ) ) ) ).
% insert_relcomp_fold
thf(fact_5576_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B,P2: B > $o,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs2 ) )
=> ( ( P2 @ X )
=> ( ( filter2 @ B @ P2 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs2 ) )
= ( linorder_insort_key @ B @ A @ F3 @ X @ ( filter2 @ B @ P2 @ Xs2 ) ) ) ) ) ) ).
% filter_insort
thf(fact_5577_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X8: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S3 ) @ X8 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z4: B,A15: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z4 ) @ A15 )
@ A15 ) )
@ X8
@ S3 ) ) ) ).
% insert_relcomp_union_fold
thf(fact_5578_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( ord_max @ A @ X4 ) @ Y4 ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_5579_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_5580_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] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A6 ) ) ) ).
% Id_on_fold
thf(fact_5581_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A8: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A8 ) ) ) ) ).
% Id_on_def
thf(fact_5582_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_5583_Id__on__iff,axiom,
! [A: $tType,X: A,Y: A,A6: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id_on @ A @ A6 ) )
= ( ( X = Y )
& ( member @ A @ X @ A6 ) ) ) ).
% Id_on_iff
thf(fact_5584_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_5585_Id__onE,axiom,
! [A: $tType,C3: product_prod @ A @ A,A6: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A6 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( C3
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_5586_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_5587_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B,X: 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 @ X @ Xs2 ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_5588_length__remove1,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs2 ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% length_remove1
thf(fact_5589_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_5590_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_5591_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,Y4: A,A8: 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,A15: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y4 = W3 ) @ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X4 @ Z4 ) @ A15 ) @ A15 ) )
@ A8
@ S3 ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_5592_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( sup_sup @ A @ X4 ) @ Y4 ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_5593_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( inf_inf @ A @ X4 ) @ Y4 ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_5594_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_5595_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_5596_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_5597_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_5598_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_5599_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( ord_less_eq @ A @ A5 @ X ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_5600_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X )
=> ! [A9: A] :
( ( member @ A @ A9 @ A6 )
=> ( ord_less_eq @ A @ A9 @ X ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_5601_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( ord_less_eq @ A @ X @ A5 ) )
=> ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_5602_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
=> ! [A9: A] :
( ( member @ A @ A9 @ A6 )
=> ( ord_less_eq @ A @ X @ A9 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_5603_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ B6 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_5604_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B6 ) @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_5605_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B6: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X4: B,Z4: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y4: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y4 ) )
@ Z4
@ B6 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_5606_relpow__finite__bounded1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),K2: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K2 @ R2 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R2 )
@ ( collect @ nat
@ ^ [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( ord_less_eq @ nat @ N5 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_5607_Rats__eq__int__div__nat,axiom,
( ( field_char_0_Rats @ real )
= ( collect @ real
@ ^ [Uu3: real] :
? [I: int,N5: nat] :
( ( Uu3
= ( divide_divide @ real @ ( ring_1_of_int @ real @ I ) @ ( semiring_1_of_nat @ real @ N5 ) ) )
& ( N5
!= ( zero_zero @ nat ) ) ) ) ) ).
% Rats_eq_int_div_nat
thf(fact_5608_lim__at__infinity__0,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,L: A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) )
= ( filterlim @ A @ A @ ( comp @ A @ A @ A @ F3 @ ( inverse_inverse @ A ) ) @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% lim_at_infinity_0
thf(fact_5609_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_5610_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_5611_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_5612_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_5613_length__filter__map,axiom,
! [A: $tType,B: $tType,P2: A > $o,F3: B > A,Xs2: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ ( map @ B @ A @ F3 @ Xs2 ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P2 @ F3 ) @ Xs2 ) ) ) ).
% length_filter_map
thf(fact_5614_finite__relpow,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),N: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% finite_relpow
thf(fact_5615_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_5616_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_5617_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_5618_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F3: C > B,G3: D > C,X: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apsnd @ D @ C @ A @ G3 @ X ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F3 @ G3 ) @ X ) ) ).
% apsnd_compose
thf(fact_5619_Rats__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( field_char_0_Rats @ A ) )
=> ( ( member @ A @ B2 @ ( field_char_0_Rats @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ).
% Rats_mult
thf(fact_5620_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_5621_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_5622_Rats__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_0
thf(fact_5623_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > C,X: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_apfst @ D @ C @ B @ G3 @ X ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F3 @ G3 ) @ X ) ) ).
% apfst_compose
thf(fact_5624_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_5625_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_5626_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_5627_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_5628_relpow__Suc__D2_H,axiom,
! [A: $tType,N: nat,R2: set @ ( product_prod @ A @ A ),X5: A,Y6: A,Z5: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z5 ) @ R2 ) )
=> ? [W: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ W ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W @ Z5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_5629_relpow__Suc__I2,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z3: A,N: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_5630_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% relpow_Suc_E2
thf(fact_5631_relpow__Suc__D2,axiom,
! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
=> ? [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% relpow_Suc_D2
thf(fact_5632_relpow__Suc__I,axiom,
! [A: $tType,X: A,Y: A,N: nat,R2: set @ ( product_prod @ A @ A ),Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) ) ) ) ).
% relpow_Suc_I
thf(fact_5633_relpow__Suc__E,axiom,
! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 ) ) ) ).
% relpow_Suc_E
thf(fact_5634_relpow__0__E,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 ) )
=> ( X = Y ) ) ).
% relpow_0_E
thf(fact_5635_relpow__0__I,axiom,
! [A: $tType,X: A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 ) ) ).
% relpow_0_I
thf(fact_5636_relpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) @ R2 ) ) ).
% relpow.simps(2)
thf(fact_5637_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 ) )
=> ( ! [X3: B,Y3: B] :
( ( H @ ( plus_plus @ B @ X3 @ Y3 ) )
= ( plus_plus @ A @ ( H @ X3 ) @ ( H @ Y3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H @ G3 ) @ A6 )
= ( H @ ( groups7311177749621191930dd_sum @ C @ B @ G3 @ A6 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_5638_relpowp__relpow__eq,axiom,
! [A: $tType,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) )
= ( ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_5639_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_5640_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_5641_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_5642_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_5643_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_5644_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_5645_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_5646_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_5647_relpow__E,axiom,
! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z3 ) )
=> ~ ! [Y3: A,M: nat] :
( ( N
= ( suc @ M ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 ) ) ) ) ) ).
% relpow_E
thf(fact_5648_relpow__E2,axiom,
! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z3 ) )
=> ~ ! [Y3: A,M: nat] :
( ( N
= ( suc @ M ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M @ R2 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_5649_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F3: D > C > A,G3: B > D,X: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F3 @ G3 ) @ X )
= ( F3 @ ( G3 @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% case_prod_comp
thf(fact_5650_relpow__empty,axiom,
! [A: $tType,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% relpow_empty
thf(fact_5651_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,H: B > C,G3: C > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ! [X3: B,Y3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ( member @ B @ Y3 @ A6 )
=> ( ( X3 != Y3 )
=> ( ( ( H @ X3 )
= ( H @ Y3 ) )
=> ( ( G3 @ ( H @ X3 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G3 @ ( image2 @ B @ C @ H @ A6 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G3 @ H ) @ A6 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_5652_DERIV__image__chain,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Da: A,G3: A > A,X: A,S2: set @ A,Db: A] :
( ( has_field_derivative @ A @ F3 @ Da @ ( topolo174197925503356063within @ A @ ( G3 @ X ) @ ( image2 @ A @ A @ G3 @ S2 ) ) )
=> ( ( has_field_derivative @ A @ G3 @ Db @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F3 @ G3 ) @ ( times_times @ A @ Da @ Db ) @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_image_chain
thf(fact_5653_DERIV__chain,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Da: A,G3: A > A,X: A,Db: A,S2: set @ A] :
( ( has_field_derivative @ A @ F3 @ Da @ ( topolo174197925503356063within @ A @ ( G3 @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ G3 @ Db @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F3 @ G3 ) @ ( times_times @ A @ Da @ Db ) @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ).
% DERIV_chain
thf(fact_5654_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_5655_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( ^ [G4: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G4 ) @ ( one_one @ A ) ) ) ) ) ).
% prod.eq_fold
thf(fact_5656_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,Y4: A] : ( product_Pair @ A @ B @ Y4 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_5657_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,Y4: B] : ( product_Pair @ B @ A @ Y4 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_5658_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B2: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
= ( ? [F4: nat > A] :
( ( ( F4 @ ( zero_zero @ nat ) )
= A3 )
& ( ( F4 @ N )
= B2 )
& ! [I: nat] :
( ( ord_less @ nat @ I @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ I ) @ ( F4 @ ( suc @ I ) ) ) @ R2 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_5659_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_5660_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_5661_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_5662_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_5663_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_5664_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_5665_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_5666_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: set @ ( set @ B ),G3: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B6 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [A16: set @ B] :
( ( member @ ( set @ B ) @ A16 @ B6 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B6 )
=> ( ( A16 != A25 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A16 )
=> ( ( member @ B @ X3 @ A25 )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( complete_Sup_Sup @ ( set @ B ) @ B6 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G3 @ B6 ) ) ) ) ) ).
% sum.Union_comp
thf(fact_5667_sum_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( ord @ B ) )
=> ! [H: nat > B,M2: nat,N: nat,G3: B > A] :
( ( bij_betw @ nat @ B @ H @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H @ M2 ) @ ( H @ N ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( set_or1337092689740270186AtMost @ B @ ( H @ M2 ) @ ( H @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G3 @ H ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ).
% sum.atLeastAtMost_reindex
thf(fact_5668_sum_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( ord @ B ) )
=> ! [H: nat > B,M2: nat,N: nat,G3: B > A] :
( ( bij_betw @ nat @ B @ H @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H @ M2 ) @ ( H @ N ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( set_or7035219750837199246ssThan @ B @ ( H @ M2 ) @ ( H @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G3 @ H ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ).
% sum.atLeastLessThan_reindex
thf(fact_5669_prod_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( ord @ B ) )
=> ! [H: nat > B,M2: nat,N: nat,G3: B > A] :
( ( bij_betw @ nat @ B @ H @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H @ M2 ) @ ( H @ N ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( set_or1337092689740270186AtMost @ B @ ( H @ M2 ) @ ( H @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G3 @ H ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ).
% prod.atLeastAtMost_reindex
thf(fact_5670_prod_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( ord @ B ) )
=> ! [H: nat > B,M2: nat,N: nat,G3: B > A] :
( ( bij_betw @ nat @ B @ H @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H @ M2 ) @ ( H @ N ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( set_or7035219750837199246ssThan @ B @ ( H @ M2 ) @ ( H @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G3 @ H ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ).
% prod.atLeastLessThan_reindex
thf(fact_5671_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
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( 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_5672_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
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( 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_5673_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
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( 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_5674_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
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( 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_5675_relpow__finite__bounded,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),K2: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K2 @ R2 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R2 )
@ ( collect @ nat
@ ^ [N5: nat] : ( ord_less_eq @ nat @ N5 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_5676_sum_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: int > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G3 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G3 @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_int_atMost_int_shift
thf(fact_5677_prod_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: int > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G3 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G3 @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_int_atMost_int_shift
thf(fact_5678_sum_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: int > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G3 @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G3 @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_int_lessThan_int_shift
thf(fact_5679_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_5680_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_5681_prod_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: int > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G3 @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G3 @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_int_lessThan_int_shift
thf(fact_5682_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N5: nat,R6: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [I: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I @ R6 )
@ ( collect @ nat
@ ^ [I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
& ( ord_less_eq @ nat @ I @ ( suc @ N5 ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_5683_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L2
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( sgn_sgn @ code_integer @ K3 )
= ( sgn_sgn @ code_integer @ L2 ) )
@ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R: code_integer,S6: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S6
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_5684_trancl__finite__eq__relpow,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( transitive_trancl @ A @ R2 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R2 )
@ ( collect @ nat
@ ^ [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( ord_less_eq @ nat @ N5 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_5685_ntrancl__Zero,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( zero_zero @ nat ) @ R2 )
= R2 ) ).
% ntrancl_Zero
thf(fact_5686_converse__trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ B2 ) @ R3 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: A,Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A3 ) ) ) ) ).
% converse_trancl_induct
thf(fact_5687_trancl__trans__induct,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),P2: A > A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( P2 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A,Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( P2 @ X3 @ Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( P2 @ Y3 @ Z )
=> ( P2 @ X3 @ Z ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_5688_trancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_5689_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_5690_irrefl__trancl__rD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_5691_converse__tranclE,axiom,
! [A: $tType,X: A,Z3: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ R3 )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ) ).
% converse_tranclE
thf(fact_5692_r__r__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 ) @ 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 ) ) ) ) ).
% r_r_into_trancl
thf(fact_5693_trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y3 ) @ R3 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: A,Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R3 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_5694_trancl__trans,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_trans
thf(fact_5695_tranclE,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ~ ! [C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C2 @ B2 ) @ R3 ) ) ) ) ).
% tranclE
thf(fact_5696_trancl_Or__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ).
% trancl.r_into_trancl
thf(fact_5697_trancl_Osimps,axiom,
! [A: $tType,A12: A,A23: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_trancl @ A @ R3 ) )
= ( ? [A7: A,B5: A] :
( ( A12 = A7 )
& ( A23 = B5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) )
| ? [A7: A,B5: A,C4: A] :
( ( A12 = A7 )
& ( A23 = C4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ ( transitive_trancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C4 ) @ R3 ) ) ) ) ).
% trancl.simps
thf(fact_5698_trancl_Ocases,axiom,
! [A: $tType,A12: A,A23: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ R3 )
=> ~ ! [B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B4 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A23 ) @ R3 ) ) ) ) ).
% trancl.cases
thf(fact_5699_trancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P2: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ! [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 ) ) @ R3 )
=> ( P2 @ 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 ) @ R3 ) )
=> ( ( 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 ) ) @ R3 )
=> ( ( P2 @ A5 @ B4 )
=> ( P2 @ Aa2 @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_5700_trancl__power,axiom,
! [A: $tType,P: product_prod @ A @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P @ ( transitive_trancl @ A @ R2 ) )
= ( ? [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( member @ ( product_prod @ A @ A ) @ P @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R2 ) ) ) ) ) ).
% trancl_power
thf(fact_5701_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu3: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_5702_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_5703_trancl__insert2,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y4: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ A3 ) @ ( transitive_trancl @ A @ R3 ) )
| ( X4 = A3 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y4 ) @ ( transitive_trancl @ A @ R3 ) )
| ( Y4 = B2 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_5704_rtrancl__finite__eq__relpow,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( transitive_rtrancl @ A @ R2 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R2 )
@ ( collect @ nat
@ ^ [N5: nat] : ( ord_less_eq @ nat @ N5 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ).
% rtrancl_finite_eq_relpow
thf(fact_5705_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F3: A > B,Ks: list @ A] :
( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( F3 @ K3 ) )
@ Ks ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F3 ) @ ( set2 @ A @ Ks ) ) ) ).
% map_of_map_restrict
thf(fact_5706_ntrancl__Suc,axiom,
! [A: $tType,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( suc @ N ) @ R2 )
= ( relcomp @ A @ A @ A @ ( transitive_ntrancl @ A @ N @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ R2 ) ) ) ).
% ntrancl_Suc
thf(fact_5707_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_5708_IdI,axiom,
! [A: $tType,A3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_5709_trancl__rtrancl__trancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_5710_rtrancl__trancl__trancl,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_5711_rtrancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_5712_rtrancl__into__trancl1,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_5713_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_5714_trancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ).
% trancl_into_rtrancl
thf(fact_5715_tranclD2,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ? [Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ R2 ) ) ) ).
% tranclD2
thf(fact_5716_rtranclD,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 )
| ( ( A3 != B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ) ).
% rtranclD
thf(fact_5717_tranclD,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ? [Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% tranclD
thf(fact_5718_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_5719_rtrancl_Ocases,axiom,
! [A: $tType,A12: A,A23: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( A23 != A12 )
=> ~ ! [B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B4 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A23 ) @ R3 ) ) ) ) ).
% rtrancl.cases
thf(fact_5720_rtrancl_Osimps,axiom,
! [A: $tType,A12: A,A23: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_rtrancl @ A @ R3 ) )
= ( ? [A7: A] :
( ( A12 = A7 )
& ( A23 = A7 ) )
| ? [A7: A,B5: A,C4: A] :
( ( A12 = A7 )
& ( A23 = C4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ ( transitive_rtrancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C4 ) @ R3 ) ) ) ) ).
% rtrancl.simps
thf(fact_5721_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A3: A,R3: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_5722_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_5723_rtranclE,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( A3 != B2 )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ B2 ) @ R3 ) ) ) ) ).
% rtranclE
thf(fact_5724_rtrancl__trans,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% rtrancl_trans
thf(fact_5725_rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( P2 @ A3 )
=> ( ! [Y3: A,Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R3 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_5726_converse__rtranclE,axiom,
! [A: $tType,X: A,Z3: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( X != Z3 )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_5727_converse__rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( P2 @ B2 )
=> ( ! [Y3: A,Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( P2 @ Z )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A3 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_5728_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_5729_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_5730_IdE,axiom,
! [A: $tType,P: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P @ ( id2 @ A ) )
=> ~ ! [X3: A] :
( P
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_5731_rtrancl__Un__separatorE,axiom,
! [A: $tType,A3: A,B2: A,P2: 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 ) ) @ P2 @ Q ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X3 ) @ ( transitive_rtrancl @ A @ P2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ Q )
=> ( X3 = Y3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_5732_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A3: A,B2: A,P2: 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 ) ) @ P2 @ Q ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B2 ) @ ( transitive_rtrancl @ A @ P2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ Q )
=> ( Y3 = X3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_5733_converse__rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P2: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( P2 @ 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 ) ) @ R3 )
=> ( ( 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 ) @ R3 ) )
=> ( ( P2 @ Aa2 @ Ba )
=> ( P2 @ A5 @ B4 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_5734_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa2: A,Xb: B,Za: A,Zb: B,R3: 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 ) @ R3 ) )
=> ( ( ( 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 ) ) @ R3 )
=> ~ ( 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 ) @ R3 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_5735_rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P2: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( P2 @ 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 ) @ R3 ) )
=> ( ( 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 ) ) @ R3 )
=> ( ( P2 @ A5 @ B4 )
=> ( P2 @ Aa2 @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_5736_relpow_Osimps_I1_J,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 )
= ( id2 @ A ) ) ).
% relpow.simps(1)
thf(fact_5737_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_5738_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 ) ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A6 )
& ( ( M2 @ X3 )
= ( some @ A @ Y ) ) ) ) ).
% ran_restrictD
thf(fact_5739_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) )
=> ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_5740_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_5741_reflcl__set__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 )
@ ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_5742_rtrancl__insert,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ A3 ) @ ( transitive_rtrancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y4 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_5743_trancl__insert,axiom,
! [A: $tType,Y: A,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A7: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B5 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_5744_restrict__map__upds,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,D5: set @ A,M2: A > ( option @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ D5 )
=> ( ( restrict_map @ A @ B @ ( map_upds @ A @ B @ M2 @ Xs2 @ Ys ) @ D5 )
= ( map_upds @ A @ B @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D5 @ ( set2 @ A @ Xs2 ) ) ) @ Xs2 @ Ys ) ) ) ) ).
% restrict_map_upds
thf(fact_5745_dual__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( max @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 ) )
= ( ord_min @ A ) ) ) ).
% dual_max
thf(fact_5746_total__on__singleton,axiom,
! [A: $tType,X: A] : ( total_on @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_5747_fun__upds__append__drop,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,M2: A > ( option @ B ),Zs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs2 @ Zs ) @ Ys )
= ( map_upds @ A @ B @ M2 @ Xs2 @ Ys ) ) ) ).
% fun_upds_append_drop
thf(fact_5748_fun__upds__append2__drop,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,M2: A > ( option @ B ),Zs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ Xs2 @ ( append @ B @ Ys @ Zs ) )
= ( map_upds @ A @ B @ M2 @ Xs2 @ Ys ) ) ) ).
% fun_upds_append2_drop
thf(fact_5749_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,I2: nat,M2: A > ( option @ B ),Ys: list @ B,Y: B] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ I2 )
=> ( ( map_upds @ A @ B @ M2 @ Xs2 @ ( list_update @ B @ Ys @ I2 @ Y ) )
= ( map_upds @ A @ B @ M2 @ Xs2 @ Ys ) ) ) ).
% map_upds_list_update2_drop
thf(fact_5750_total__onI,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ( member @ A @ Y3 @ A6 )
=> ( ( X3 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) )
=> ( total_on @ A @ A6 @ R3 ) ) ).
% total_onI
thf(fact_5751_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A8: set @ A,R: set @ ( product_prod @ A @ A )] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ A8 )
=> ( ( X4 != Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_5752_ord_Omax_Ocong,axiom,
! [A: $tType] :
( ( max @ A )
= ( max @ A ) ) ).
% ord.max.cong
thf(fact_5753_ord_Omax__def,axiom,
! [A: $tType] :
( ( max @ A )
= ( ^ [Less_eq: A > A > $o,A7: A,B5: A] : ( if @ A @ ( Less_eq @ A7 @ B5 ) @ B5 @ A7 ) ) ) ).
% ord.max_def
thf(fact_5754_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys: list @ B,M2: A > ( option @ B ),X: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ Xs2 @ Ys ) @ X @ ( some @ B @ ( nth @ B @ Ys @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_5755_bezw__aux,axiom,
! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X @ Y ) )
= ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X @ Y ) ) @ ( semiring_1_of_nat @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X @ Y ) ) @ ( semiring_1_of_nat @ int @ Y ) ) ) ) ).
% bezw_aux
thf(fact_5756_Field__insert,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) )
= ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) ) ) ).
% Field_insert
thf(fact_5757_gcd__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% gcd_eq_0_iff
thf(fact_5758_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X )
= X ) ).
% gcd_0_left_nat
thf(fact_5759_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ X @ ( zero_zero @ nat ) )
= X ) ).
% gcd_0_nat
thf(fact_5760_gcd__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( gcd_gcd @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% gcd_nat.right_neutral
thf(fact_5761_gcd__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( gcd_gcd @ nat @ A3 @ B2 ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.neutr_eq_iff
thf(fact_5762_gcd__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% gcd_nat.left_neutral
thf(fact_5763_gcd__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( gcd_gcd @ nat @ A3 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.eq_neutr_iff
thf(fact_5764_gcd__Suc__0,axiom,
! [M2: nat] :
( ( gcd_gcd @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% gcd_Suc_0
thf(fact_5765_gcd__pos__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_gcd @ nat @ M2 @ N ) )
= ( ( M2
!= ( zero_zero @ nat ) )
| ( N
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_pos_nat
thf(fact_5766_image__map__upd,axiom,
! [B: $tType,A: $tType,X: A,A6: set @ A,M2: A > ( option @ B ),Y: B] :
( ~ ( member @ A @ X @ A6 )
=> ( ( image2 @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y ) ) @ A6 )
= ( image2 @ A @ ( option @ B ) @ M2 @ A6 ) ) ) ).
% image_map_upd
thf(fact_5767_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_5768_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_5769_ran__map__upd,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
( ( ( M2 @ A3 )
= ( none @ A ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ A3 @ ( some @ A @ B2 ) ) )
= ( insert @ A @ B2 @ ( ran @ B @ A @ M2 ) ) ) ) ).
% ran_map_upd
thf(fact_5770_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_5771_map__upd__Some__unfold,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),A3: B,B2: A,X: B,Y: A] :
( ( ( fun_upd @ B @ ( option @ A ) @ M2 @ A3 @ ( some @ A @ B2 ) @ X )
= ( some @ A @ Y ) )
= ( ( ( X = A3 )
& ( B2 = Y ) )
| ( ( X != A3 )
& ( ( M2 @ X )
= ( some @ A @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_5772_map__upd__triv,axiom,
! [A: $tType,B: $tType,T2: B > ( option @ A ),K2: B,X: A] :
( ( ( T2 @ K2 )
= ( some @ A @ X ) )
=> ( ( fun_upd @ B @ ( option @ A ) @ T2 @ K2 @ ( some @ A @ X ) )
= T2 ) ) ).
% map_upd_triv
thf(fact_5773_map__upd__eqD1,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),A3: A,X: B,N: A > ( option @ B ),Y: B] :
( ( ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ X ) )
= ( fun_upd @ A @ ( option @ B ) @ N @ A3 @ ( some @ B @ Y ) ) )
=> ( X = Y ) ) ).
% map_upd_eqD1
thf(fact_5774_map__upd__nonempty,axiom,
! [B: $tType,A: $tType,T2: A > ( option @ B ),K2: A,X: B] :
( ( fun_upd @ A @ ( option @ B ) @ T2 @ K2 @ ( some @ B @ X ) )
!= ( ^ [X4: A] : ( none @ B ) ) ) ).
% map_upd_nonempty
thf(fact_5775_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_5776_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_5777_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_5778_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_5779_gcd__dvd__prod,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,K2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( times_times @ A @ K2 @ B2 ) ) ) ).
% gcd_dvd_prod
thf(fact_5780_gcd__mult__distrib__nat,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( times_times @ nat @ K2 @ ( gcd_gcd @ nat @ M2 @ N ) )
= ( gcd_gcd @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) ) ) ).
% gcd_mult_distrib_nat
thf(fact_5781_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_5782_FieldI1,axiom,
! [A: $tType,I2: A,J2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J2 ) @ R2 )
=> ( member @ A @ I2 @ ( field2 @ A @ R2 ) ) ) ).
% FieldI1
thf(fact_5783_FieldI2,axiom,
! [A: $tType,I2: A,J2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J2 ) @ R2 )
=> ( member @ A @ J2 @ ( field2 @ A @ R2 ) ) ) ).
% FieldI2
thf(fact_5784_gcd__non__0__nat,axiom,
! [Y: nat,X: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ( ( gcd_gcd @ nat @ X @ Y )
= ( gcd_gcd @ nat @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) ) ).
% gcd_non_0_nat
thf(fact_5785_gcd__nat_Osimps,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X4: nat,Y4: nat] :
( if @ nat
@ ( Y4
= ( zero_zero @ nat ) )
@ X4
@ ( gcd_gcd @ nat @ Y4 @ ( modulo_modulo @ nat @ X4 @ Y4 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_5786_gcd__nat_Oelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y = X ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_5787_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C3 @ A3 ) )
= ( gcd_gcd @ A @ B2 @ C3 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_5788_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A3 ) @ C3 )
= ( gcd_gcd @ A @ B2 @ C3 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_5789_bezout__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [X3: nat,Y3: nat] :
( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% bezout_nat
thf(fact_5790_bezout__gcd__nat_H,axiom,
! [B2: nat,A3: nat] :
? [X3: nat,Y3: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ ( times_times @ nat @ A3 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X3 ) @ ( times_times @ nat @ B2 @ Y3 ) )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A3 @ Y3 ) @ ( times_times @ nat @ B2 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A3 @ Y3 ) )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_5791_Field__natLeq__on,axiom,
! [N: nat] :
( ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y4: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y4 @ N )
& ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) )
= ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_5792_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M6: nat,N5: nat] :
( ( dvd_dvd @ nat @ M6 @ N5 )
& ( M6 != N5 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_5793_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_5794_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( gcd_gcd @ nat @ M2 @ N )
= ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D4: nat] :
( ( dvd_dvd @ nat @ D4 @ M2 )
& ( dvd_dvd @ nat @ D4 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_5795_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( ^ [A8: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A8 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A8 ) @ ( one_one @ A ) ) ) ) ) ).
% Gcd_fin.eq_fold
thf(fact_5796_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys: list @ B,Xs2: list @ A,Zs: list @ B,X: A,Y: B,Z3: B] :
( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Zs )
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) ) @ X @ ( some @ B @ Y ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Zs ) ) @ X @ ( some @ B @ Z3 ) ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys ) )
= ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Zs ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_5797_map__upd__upds__conv__if,axiom,
! [A: $tType,B: $tType,X: A,Ys: list @ B,Xs2: list @ A,F3: A > ( option @ B ),Y: B] :
( ( ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs2 ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ ( some @ B @ Y ) ) @ Xs2 @ Ys )
= ( map_upds @ A @ B @ F3 @ Xs2 @ Ys ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs2 ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ ( some @ B @ Y ) ) @ Xs2 @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F3 @ Xs2 @ Ys ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_5798_map__of_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Ps: list @ ( product_prod @ A @ B )] :
( ( map_of @ A @ B @ ( cons @ ( product_prod @ A @ B ) @ P @ Ps ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Ps ) @ ( product_fst @ A @ B @ P ) @ ( some @ B @ ( product_snd @ A @ B @ P ) ) ) ) ).
% map_of.simps(2)
thf(fact_5799_Total__subset__Id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) )
=> ( ( R3
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
| ? [A5: A] :
( R3
= ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_5800_gcd__nat_Opelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y = X ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_5801_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),X: B,Y: A,Z3: A] :
( ( ( M2 @ X )
= ( some @ A @ Y ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M2 @ ( dom @ B @ A @ M2 ) )
=> ( ~ ( member @ A @ Z3 @ ( ran @ B @ A @ M2 ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ X @ ( some @ A @ Z3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M2 ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Z3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_5802_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_5803_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_5804_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_5805_gcd__mult__distrib__int,axiom,
! [K2: int,M2: int,N: int] :
( ( times_times @ int @ ( abs_abs @ int @ K2 ) @ ( gcd_gcd @ int @ M2 @ N ) )
= ( gcd_gcd @ int @ ( times_times @ int @ K2 @ M2 ) @ ( times_times @ int @ K2 @ N ) ) ) ).
% gcd_mult_distrib_int
thf(fact_5806_bezout__int,axiom,
! [X: int,Y: int] :
? [U5: int,V2: int] :
( ( plus_plus @ int @ ( times_times @ int @ U5 @ X ) @ ( times_times @ int @ V2 @ Y ) )
= ( gcd_gcd @ int @ X @ Y ) ) ).
% bezout_int
thf(fact_5807_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_5808_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_5809_insert__dom,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),X: B,Y: A] :
( ( ( F3 @ X )
= ( some @ A @ Y ) )
=> ( ( insert @ B @ X @ ( dom @ B @ A @ F3 ) )
= ( dom @ B @ A @ F3 ) ) ) ).
% insert_dom
thf(fact_5810_finite__Map__induct,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),P2: ( A > ( option @ B ) ) > $o] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) )
=> ( ( P2
@ ^ [X4: A] : ( none @ B ) )
=> ( ! [K: A,V2: B,M: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M ) )
=> ( ~ ( member @ A @ K @ ( dom @ A @ B @ M ) )
=> ( ( P2 @ M )
=> ( P2 @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V2 ) ) ) ) ) )
=> ( P2 @ M2 ) ) ) ) ).
% finite_Map_induct
thf(fact_5811_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),X: A] :
( ( ( dom @ A @ B @ F3 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V4: B] :
( F3
= ( fun_upd @ A @ ( option @ B )
@ ^ [X4: A] : ( none @ B )
@ X
@ ( some @ B @ V4 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_5812_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_5813_UnderS__def,axiom,
! [A: $tType] :
( ( order_UnderS @ A )
= ( ^ [R: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ( B5 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X4 ) @ R ) ) ) ) ) ) ) ).
% UnderS_def
thf(fact_5814_Under__def,axiom,
! [A: $tType] :
( ( order_Under @ A )
= ( ^ [R: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X4 ) @ R ) ) ) ) ) ) ).
% Under_def
thf(fact_5815_Above__def,axiom,
! [A: $tType] :
( ( order_Above @ A )
= ( ^ [R: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ B5 ) @ R ) ) ) ) ) ) ).
% Above_def
thf(fact_5816_cofinal__def,axiom,
! [A: $tType] :
( ( bNF_Ca7293521722713021262ofinal @ A )
= ( ^ [A8: set @ A,R: set @ ( product_prod @ A @ A )] :
! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R ) )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A8 )
& ( X4 != Y4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R ) ) ) ) ) ).
% cofinal_def
thf(fact_5817_graph__map__upd,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),K2: A,V3: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K2 @ ( some @ B @ V3 ) ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V3 ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K2 @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_5818_linear__order__on__singleton,axiom,
! [A: $tType,X: A] : ( order_679001287576687338der_on @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_5819_in__graphD,axiom,
! [A: $tType,B: $tType,K2: A,V3: B,M2: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V3 ) @ ( graph @ A @ B @ M2 ) )
=> ( ( M2 @ K2 )
= ( some @ B @ V3 ) ) ) ).
% in_graphD
thf(fact_5820_in__graphI,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),K2: B,V3: A] :
( ( ( M2 @ K2 )
= ( some @ A @ V3 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K2 @ V3 ) @ ( graph @ B @ A @ M2 ) ) ) ).
% in_graphI
thf(fact_5821_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K2: A,V3: B,M2: A > ( option @ B ),A6: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V3 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A6 ) ) )
=> ( member @ A @ K2 @ A6 ) ) ).
% graph_restrictD(1)
thf(fact_5822_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K2: A,V3: B,M2: A > ( option @ B ),A6: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V3 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A6 ) ) )
=> ( ( M2 @ K2 )
= ( some @ B @ V3 ) ) ) ).
% graph_restrictD(2)
thf(fact_5823_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M6: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A7: A,B5: B] :
( ( Uu3
= ( product_Pair @ A @ B @ A7 @ B5 ) )
& ( ( M6 @ A7 )
= ( some @ B @ B5 ) ) ) ) ) ) ).
% graph_def
thf(fact_5824_Linear__order__in__diff__Id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_5825_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M6: A > ( option @ B )] :
( image2 @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ ( the2 @ B @ ( M6 @ X4 ) ) )
@ ( dom @ A @ B @ M6 ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_5826_map__upds__fold__map__upd,axiom,
! [B: $tType,A: $tType] :
( ( map_upds @ A @ B )
= ( ^ [M6: A > ( option @ B ),Ks2: list @ A,Vs3: list @ B] :
( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B )
@ ^ [N5: A > ( option @ B )] :
( product_case_prod @ A @ B @ ( A > ( option @ B ) )
@ ^ [K3: A,V4: B] : ( fun_upd @ A @ ( option @ B ) @ N5 @ K3 @ ( some @ B @ V4 ) ) )
@ M6
@ ( zip @ A @ B @ Ks2 @ Vs3 ) ) ) ) ).
% map_upds_fold_map_upd
thf(fact_5827_increasing__Bseq__subseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G3: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X3 @ Y3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X3 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ Y3 ) ) ) )
=> ( ( 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_5828_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_5829_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_5830_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_5831_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% strict_mono_less_eq
thf(fact_5832_strict__mono__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [R3: A > B,M2: A,N: A] :
( ( order_strict_mono @ A @ B @ R3 )
=> ( ( ord_less_eq @ A @ M2 @ N )
=> ( ord_less_eq @ B @ ( R3 @ M2 ) @ ( R3 @ N ) ) ) ) ) ).
% strict_mono_leD
thf(fact_5833_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less @ A @ X @ Y )
=> ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% strict_monoD
thf(fact_5834_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( order_strict_mono @ A @ B @ F3 ) ) ) ).
% strict_monoI
thf(fact_5835_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F4: A > B] :
! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ B @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_5836_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% strict_mono_less
thf(fact_5837_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ) ).
% strict_mono_eq
thf(fact_5838_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_5839_strict__mono__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_strict_mono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N5: nat] : ( ord_less @ A @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) ) ) ) ) ).
% strict_mono_Suc_iff
thf(fact_5840_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_5841_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_5842_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_5843_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_5844_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_5845_butlast__list__update,axiom,
! [A: $tType,K2: nat,Xs2: list @ A,X: A] :
( ( ( K2
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs2 @ K2 @ X ) )
= ( butlast @ A @ Xs2 ) ) )
& ( ( K2
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs2 @ K2 @ X ) )
= ( list_update @ A @ ( butlast @ A @ Xs2 ) @ K2 @ X ) ) ) ) ).
% butlast_list_update
thf(fact_5846_summable__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G3: nat > nat,F3: nat > A] :
( ( order_strict_mono @ nat @ nat @ G3 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image2 @ nat @ nat @ G3 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( summable @ A
@ ^ [N5: nat] : ( F3 @ ( G3 @ N5 ) ) )
= ( summable @ A @ F3 ) ) ) ) ) ).
% summable_mono_reindex
thf(fact_5847_sums__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G3: nat > nat,F3: nat > A,C3: A] :
( ( order_strict_mono @ nat @ nat @ G3 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image2 @ nat @ nat @ G3 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [N5: nat] : ( F3 @ ( G3 @ N5 ) )
@ C3 )
= ( sums @ A @ F3 @ C3 ) ) ) ) ) ).
% sums_mono_reindex
thf(fact_5848_suminf__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G3: nat > nat,F3: nat > A] :
( ( order_strict_mono @ nat @ nat @ G3 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image2 @ nat @ nat @ G3 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A
@ ^ [N5: nat] : ( F3 @ ( G3 @ N5 ) ) )
= ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_mono_reindex
thf(fact_5849_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
= ( ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R3 ) )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ! [Y4: A] :
( ( member @ A @ Y4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_5850_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
@ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
@ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
@ ^ [A13: A,A24: A] :
( product_case_prod @ A @ A @ $o
@ ^ [B1: A,B22: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A13 @ ( insert @ A @ A24 @ ( insert @ A @ B1 @ ( insert @ A @ B22 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R ) )
& ( ( ( A13 = B1 )
& ( A24 = B22 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A13 @ A24 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R @ B1 @ B22 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A13 @ A24 )
= ( bNF_We1388413361240627857o_max2 @ A @ R @ B1 @ B22 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A13 @ B1 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A13 @ A24 )
= ( bNF_We1388413361240627857o_max2 @ A @ R @ B1 @ B22 ) )
& ( A13 = B1 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A24 @ B22 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_5851_DeMoivre2,axiom,
! [R3: real,A3: real,N: nat] :
( ( power_power @ complex @ ( rcis @ R3 @ A3 ) @ N )
= ( rcis @ ( power_power @ real @ R3 @ N ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A3 ) ) ) ).
% DeMoivre2
thf(fact_5852_wf__insert,axiom,
! [A: $tType,Y: A,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R3 ) )
= ( ( wf @ A @ R3 )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% wf_insert
thf(fact_5853_Re__rcis,axiom,
! [R3: real,A3: real] :
( ( re @ ( rcis @ R3 @ A3 ) )
= ( times_times @ real @ R3 @ ( cos @ real @ A3 ) ) ) ).
% Re_rcis
thf(fact_5854_Im__rcis,axiom,
! [R3: real,A3: real] :
( ( im @ ( rcis @ R3 @ A3 ) )
= ( times_times @ real @ R3 @ ( sin @ real @ A3 ) ) ) ).
% Im_rcis
thf(fact_5855_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
~ ? [F4: nat > A] :
! [I: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ ( suc @ I ) ) @ ( F4 @ I ) ) @ R ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_5856_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),F3: nat > A] :
( ( wf @ A @ R3 )
=> ~ ! [K: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F3 @ ( suc @ K ) ) @ ( F3 @ K ) ) @ R3 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_5857_wf__induct__rule,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P2: A > $o,A3: A] :
( ( wf @ A @ R3 )
=> ( ! [X3: A] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R3 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A3 ) ) ) ).
% wf_induct_rule
thf(fact_5858_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [Q6: set @ A] :
( ? [X4: A] : ( member @ A @ X4 @ Q6 )
=> ? [X4: A] :
( ( member @ A @ X4 @ Q6 )
& ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R )
=> ~ ( member @ A @ Y4 @ Q6 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_5859_wf__not__refl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R3 ) ) ).
% wf_not_refl
thf(fact_5860_wf__not__sym,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,X: A] :
( ( wf @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R3 ) ) ) ).
% wf_not_sym
thf(fact_5861_wf__irrefl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R3 ) ) ).
% wf_irrefl
thf(fact_5862_wf__induct,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P2: A > $o,A3: A] :
( ( wf @ A @ R3 )
=> ( ! [X3: A] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R3 )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A3 ) ) ) ).
% wf_induct
thf(fact_5863_wf__asym,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,X: A] :
( ( wf @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R3 ) ) ) ).
% wf_asym
thf(fact_5864_wfUNIVI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [P8: A > $o,X3: A] :
( ! [Xa: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Xa ) @ R3 )
=> ( P8 @ Y3 ) )
=> ( P8 @ Xa ) )
=> ( P8 @ X3 ) )
=> ( wf @ A @ R3 ) ) ).
% wfUNIVI
thf(fact_5865_wfI__min,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Q7: set @ A] :
( ( member @ A @ X3 @ Q7 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Q7 )
& ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Xa ) @ R2 )
=> ~ ( member @ A @ Y3 @ Q7 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wfI_min
thf(fact_5866_wfE__min,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Q: set @ A] :
( ( wf @ A @ R2 )
=> ( ( member @ A @ X @ Q )
=> ~ ! [Z: A] :
( ( member @ A @ Z @ Q )
=> ~ ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z ) @ R2 )
=> ~ ( member @ A @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_5867_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [P4: A > $o] :
( ! [X4: A] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R )
=> ( P4 @ Y4 ) )
=> ( P4 @ X4 ) )
=> ! [X7: A] : ( P4 @ X7 ) ) ) ) ).
% wf_def
thf(fact_5868_wfE__min_H,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Q: set @ A] :
( ( wf @ A @ R2 )
=> ( ( Q
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [Z: A] :
( ( member @ A @ Z @ Q )
=> ~ ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z ) @ R2 )
=> ~ ( member @ A @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_5869_wf__bounded__measure,axiom,
! [A: $tType,R3: 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 ) @ R3 )
=> ( ( 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 @ R3 ) ) ).
% wf_bounded_measure
thf(fact_5870_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),P2: B > $o,K2: B,M2: B > A] :
( ( wf @ A @ R3 )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) )
=> ( ( P2 @ K2 )
=> ? [X3: B] :
( ( P2 @ X3 )
& ! [Y6: B] :
( ( P2 @ Y6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M2 @ X3 ) @ ( M2 @ Y6 ) ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_5871_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [A8: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R ) )
& ( A8
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ! [Y4: A] :
( ( member @ A @ Y4 @ A8 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_5872_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R3: 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 ) @ R3 )
=> ( ( 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 @ R3 ) ) ).
% wf_bounded_set
thf(fact_5873_rcis__mult,axiom,
! [R1: real,A3: real,R22: real,B2: real] :
( ( times_times @ complex @ ( rcis @ R1 @ A3 ) @ ( rcis @ R22 @ B2 ) )
= ( rcis @ ( times_times @ real @ R1 @ R22 ) @ ( plus_plus @ real @ A3 @ B2 ) ) ) ).
% rcis_mult
thf(fact_5874_rcis__def,axiom,
( rcis
= ( ^ [R: real,A7: real] : ( times_times @ complex @ ( real_Vector_of_real @ complex @ R ) @ ( cis @ A7 ) ) ) ) ).
% rcis_def
thf(fact_5875_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P2: ( A > B ) > A > B > $o] :
( ( wf @ A @ R2 )
=> ( ! [F2: A > B,G2: A > B,X3: A,R4: B] :
( ! [Z5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ X3 ) @ R2 )
=> ( ( F2 @ Z5 )
= ( G2 @ Z5 ) ) )
=> ( ( P2 @ F2 @ X3 @ R4 )
= ( P2 @ G2 @ X3 @ R4 ) ) )
=> ( ! [X3: A,F2: A > B] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R2 )
=> ( P2 @ F2 @ Y6 @ ( F2 @ Y6 ) ) )
=> ? [X_1: B] : ( P2 @ F2 @ X3 @ X_1 ) )
=> ? [F2: A > B] :
! [X5: A] : ( P2 @ F2 @ X5 @ ( F2 @ X5 ) ) ) ) ) ).
% dependent_wf_choice
thf(fact_5876_lexn_Osimps_I2_J,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),N: nat] :
( ( lexn @ A @ R3 @ ( 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 ) @ R3 @ ( lexn @ A @ R3 @ 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_5877_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_5878_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [Y4: B] : Y4 )
= ( ^ [Z4: product_prod @ A @ B] : Z4 ) ) ).
% map_prod_ident
thf(fact_5879_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_5880_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ X ) )
= ( F3 @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_5881_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > B,G3: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F3 @ G3 @ X ) )
= ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_5882_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R2: set @ ( product_prod @ A @ B ),F3: A > C,G3: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R2 )
=> ( 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 ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_5883_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_5884_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_5885_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_5886_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,I2: B > D,Prod: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F @ F3 @ G3 @ ( product_map_prod @ A @ C @ B @ D @ H @ I2 @ Prod ) )
= ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ F3 @ H ) @ ( comp @ D @ F @ B @ G3 @ I2 ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_5887_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,I2: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F @ F3 @ G3 ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I2 ) )
= ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ F3 @ H ) @ ( comp @ D @ F @ B @ G3 @ I2 ) ) ) ).
% map_prod.comp
thf(fact_5888_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F3: C > A,G3: D > B,R2: 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 ) @ R2 ) )
=> ~ ! [X3: C,Y3: D] :
( ( C3
= ( product_Pair @ A @ B @ ( F3 @ X3 ) @ ( G3 @ Y3 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_5889_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,Y4: B] : ( product_Pair @ C @ D @ ( F4 @ X4 ) @ ( G4 @ Y4 ) ) ) ) ) ).
% map_prod_def
thf(fact_5890_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_5891_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A12: nat,A23: nat,A33: A,P2: ( 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 ) @ A12 @ ( product_Pair @ nat @ A @ A23 @ 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 )
=> ( P2 @ F2 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B4 @ ( F2 @ A5 @ Acc ) ) )
=> ( P2 @ F2 @ A5 @ B4 @ Acc ) ) )
=> ( P2 @ A0 @ A12 @ A23 @ A33 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_5892_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: nat > A > A,Xa2: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ 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 ) ) @ X @ ( 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 @ X @ ( plus_plus @ nat @ Xa2 @ ( one_one @ nat ) ) @ Xb @ ( X @ 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 ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa2 @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_5893_tendsto__iff__uniformity,axiom,
! [A: $tType,B: $tType] :
( ( topolo7287701948861334536_space @ B )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
= ( ! [E6: ( product_prod @ B @ B ) > $o] :
( ( eventually @ ( product_prod @ B @ B ) @ E6 @ ( topolo7806501430040627800ormity @ B ) )
=> ( eventually @ A
@ ^ [X4: A] : ( E6 @ ( product_Pair @ B @ B @ ( F3 @ X4 ) @ L ) )
@ F5 ) ) ) ) ) ).
% tendsto_iff_uniformity
thf(fact_5894_independentD,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A,T2: set @ A,U: A > real,V3: A] :
( ~ ( real_V358717886546972837endent @ A @ S2 )
=> ( ( finite_finite2 @ A @ T2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S2 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V4: A] : ( real_V8093663219630862766scaleR @ A @ ( U @ V4 ) @ V4 )
@ T2 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ V3 @ T2 )
=> ( ( U @ V3 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% independentD
thf(fact_5895_dependent__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [B8: set @ A] :
? [X7: A > real] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( X7 @ X4 )
!= ( zero_zero @ real ) ) ) )
& ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( X7 @ X4 )
!= ( zero_zero @ real ) ) )
@ B8 )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( X7 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( X7 @ X4 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
& ? [X4: A] :
( ( X7 @ X4 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% dependent_alt
thf(fact_5896_dependent__single,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V358717886546972837endent @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% dependent_single
thf(fact_5897_uniformity__transE,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ~ ! [D9: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ D9 @ ( topolo7806501430040627800ormity @ A ) )
=> ~ ! [X5: A,Y6: A] :
( ( D9 @ ( product_Pair @ A @ A @ X5 @ Y6 ) )
=> ! [Z5: A] :
( ( D9 @ ( product_Pair @ A @ A @ Y6 @ Z5 ) )
=> ( E5 @ ( product_Pair @ A @ A @ X5 @ Z5 ) ) ) ) ) ) ) ).
% uniformity_transE
thf(fact_5898_uniformity__trans,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [D9: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ D9 @ ( topolo7806501430040627800ormity @ A ) )
& ! [X5: A,Y6: A,Z5: A] :
( ( D9 @ ( product_Pair @ A @ A @ X5 @ Y6 ) )
=> ( ( D9 @ ( product_Pair @ A @ A @ Y6 @ Z5 ) )
=> ( E5 @ ( product_Pair @ A @ A @ X5 @ Z5 ) ) ) ) ) ) ) ).
% uniformity_trans
thf(fact_5899_uniformity__refl,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o,X: A] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ( E5 @ ( product_Pair @ A @ A @ X @ X ) ) ) ) ).
% uniformity_refl
thf(fact_5900_dependent__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A6: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ A6 )
=> ( real_V358717886546972837endent @ A @ A6 ) ) ) ).
% dependent_zero
thf(fact_5901_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,Y4: A] : ( E5 @ ( product_Pair @ A @ A @ Y4 @ X4 ) ) )
@ ( topolo7806501430040627800ormity @ A ) ) ) ) ).
% uniformity_sym
thf(fact_5902_Cauchy__uniform__iff,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X7: nat > A] :
! [P4: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P4 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [N6: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ N6 @ N5 )
=> ! [M6: nat] :
( ( ord_less_eq @ nat @ N6 @ M6 )
=> ( P4 @ ( product_Pair @ A @ A @ ( X7 @ N5 ) @ ( X7 @ M6 ) ) ) ) ) ) ) ) ) ).
% Cauchy_uniform_iff
thf(fact_5903_dependent__finite,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( real_V358717886546972837endent @ A @ S3 )
= ( ? [U2: A > real] :
( ? [X4: A] :
( ( member @ A @ X4 @ S3 )
& ( ( U2 @ X4 )
!= ( zero_zero @ real ) ) )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V4: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V4 ) @ V4 )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% dependent_finite
thf(fact_5904_independent__if__scalars__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [F2: A > real,X3: A] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [Y4: A] : ( real_V8093663219630862766scaleR @ A @ ( F2 @ Y4 ) @ Y4 )
@ A6 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( F2 @ X3 )
= ( zero_zero @ real ) ) ) )
=> ~ ( real_V358717886546972837endent @ A @ A6 ) ) ) ) ).
% independent_if_scalars_zero
thf(fact_5905_eventually__uniformity__metric,axiom,
! [A: $tType] :
( ( real_V768167426530841204y_dist @ A )
=> ! [P2: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P2 @ ( topolo7806501430040627800ormity @ A ) )
= ( ? [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
& ! [X4: A,Y4: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ Y4 ) @ E4 )
=> ( P2 @ ( product_Pair @ A @ A @ X4 @ Y4 ) ) ) ) ) ) ) ).
% eventually_uniformity_metric
thf(fact_5906_independent__explicit__finite__subsets,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A6: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ A6 ) )
= ( ! [S7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S7 @ A6 )
=> ( ( finite_finite2 @ A @ S7 )
=> ! [U2: A > real] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V4: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V4 ) @ V4 )
@ S7 )
= ( zero_zero @ A ) )
=> ! [X4: A] :
( ( member @ A @ X4 @ S7 )
=> ( ( U2 @ X4 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% independent_explicit_finite_subsets
thf(fact_5907_independent__explicit__module,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ S2 ) )
= ( ! [T3: set @ A,U2: A > real,V4: A] :
( ( finite_finite2 @ A @ T3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T3 @ S2 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [W3: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ W3 ) @ W3 )
@ T3 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ V4 @ T3 )
=> ( ( U2 @ V4 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% independent_explicit_module
thf(fact_5908_dependent__explicit,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [S6: set @ A] :
? [T3: set @ A] :
( ( finite_finite2 @ A @ T3 )
& ( ord_less_eq @ ( set @ A ) @ T3 @ S6 )
& ? [U2: A > real] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V4: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V4 ) @ V4 )
@ T3 )
= ( zero_zero @ A ) )
& ? [X4: A] :
( ( member @ A @ X4 @ T3 )
& ( ( U2 @ X4 )
!= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% dependent_explicit
thf(fact_5909_independentD__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B6: set @ A,X8: A > real,X: A] :
( ~ ( real_V358717886546972837endent @ A @ B6 )
=> ( ( 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 ) )
=> ( ( X8 @ X )
= ( zero_zero @ real ) ) ) ) ) ) ) ).
% independentD_alt
thf(fact_5910_independent__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B6: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ B6 ) )
= ( ! [X7: A > real] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( X7 @ X4 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( X7 @ X4 )
!= ( zero_zero @ real ) ) )
@ B6 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( X7 @ X4 ) @ X4 )
@ ( collect @ A
@ ^ [X4: A] :
( ( X7 @ X4 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
=> ! [X4: A] :
( ( X7 @ X4 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% independent_alt
thf(fact_5911_totally__bounded__def,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S7: set @ A] :
! [E6: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E6 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [X7: set @ A] :
( ( finite_finite2 @ A @ X7 )
& ! [X4: A] :
( ( member @ A @ X4 @ S7 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ X7 )
& ( E6 @ ( product_Pair @ A @ A @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ).
% totally_bounded_def
thf(fact_5912_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,Y4: A] :
( product_case_prod @ A @ A @ $o
@ ^ [Y8: A,Z4: A] :
( ( Y4 = Y8 )
=> ( 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_5913_uniformly__continuous__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo7287701948861334536_space @ A )
& ( topolo7287701948861334536_space @ B ) )
=> ! [S2: set @ A,F3: A > B,E5: ( product_prod @ B @ B ) > $o] :
( ( topolo6026614971017936543ous_on @ A @ B @ S2 @ F3 )
=> ( ( eventually @ ( product_prod @ B @ B ) @ E5 @ ( topolo7806501430040627800ormity @ B ) )
=> ( eventually @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( member @ A @ Y4 @ S2 )
=> ( E5 @ ( product_Pair @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) ) ) ) )
@ ( topolo7806501430040627800ormity @ A ) ) ) ) ) ).
% uniformly_continuous_onD
thf(fact_5914_eventually__prod__filter,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,F5: filter @ A,G7: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ P2 @ ( prod_filter @ A @ B @ F5 @ G7 ) )
= ( ? [Pf: A > $o,Pg: B > $o] :
( ( eventually @ A @ Pf @ F5 )
& ( eventually @ B @ Pg @ G7 )
& ! [X4: A,Y4: B] :
( ( Pf @ X4 )
=> ( ( Pg @ Y4 )
=> ( P2 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) ) ) ) ) ) ).
% eventually_prod_filter
thf(fact_5915_eventually__prod__same,axiom,
! [A: $tType,P2: ( product_prod @ A @ A ) > $o,F5: filter @ A] :
( ( eventually @ ( product_prod @ A @ A ) @ P2 @ ( prod_filter @ A @ A @ F5 @ F5 ) )
= ( ? [Q6: A > $o] :
( ( eventually @ A @ Q6 @ F5 )
& ! [X4: A,Y4: A] :
( ( Q6 @ X4 )
=> ( ( Q6 @ Y4 )
=> ( P2 @ ( product_Pair @ A @ A @ X4 @ Y4 ) ) ) ) ) ) ) ).
% eventually_prod_same
thf(fact_5916_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_5917_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B,G7: filter @ B,F5: filter @ A,G3: A > C,H5: filter @ C] :
( ( filterlim @ A @ B @ F3 @ G7 @ F5 )
=> ( ( filterlim @ A @ C @ G3 @ H5 @ F5 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( prod_filter @ B @ C @ G7 @ H5 )
@ F5 ) ) ) ).
% filterlim_Pair
thf(fact_5918_tendsto__mult__Pair,axiom,
! [A: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [A3: A,B2: A] :
( filterlim @ ( product_prod @ A @ A ) @ A
@ ^ [X4: product_prod @ A @ A] : ( times_times @ A @ ( product_fst @ A @ A @ X4 ) @ ( product_snd @ A @ A @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ A3 @ B2 ) )
@ ( prod_filter @ A @ A @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( topolo7230453075368039082e_nhds @ A @ B2 ) ) ) ) ).
% tendsto_mult_Pair
thf(fact_5919_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F5: filter @ A,G7: filter @ B,H5: filter @ C] :
( ( prod_filter @ ( product_prod @ A @ B ) @ C @ ( prod_filter @ A @ B @ F5 @ G7 ) @ H5 )
= ( 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 )
@ ^ [Y4: B] : ( product_Pair @ ( product_prod @ A @ B ) @ C @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) ) )
@ ( prod_filter @ A @ ( product_prod @ B @ C ) @ F5 @ ( prod_filter @ B @ C @ G7 @ H5 ) ) ) ) ).
% prod_filter_assoc
thf(fact_5920_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep: itself @ A,N5: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_5921_acyclic__insert,axiom,
! [A: $tType,Y: A,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R3 ) )
= ( ( transitive_acyclic @ A @ R3 )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% acyclic_insert
thf(fact_5922_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > A,G3: C > B,F5: filter @ C] :
( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) )
@ ( filtermap @ C @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ F5 )
@ ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F3 @ F5 ) @ ( filtermap @ C @ B @ G3 @ F5 ) ) ) ).
% filtermap_Pair
thf(fact_5923_possible__bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Ty: itself @ A] : ( bit_se6407376104438227557le_bit @ A @ Ty @ ( zero_zero @ nat ) ) ) ).
% possible_bit_0
thf(fact_5924_possible__bit__less__imp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Tyrep2: itself @ A,I2: nat,J2: nat] :
( ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ I2 )
=> ( ( ord_less_eq @ nat @ J2 @ I2 )
=> ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ J2 ) ) ) ) ).
% possible_bit_less_imp
thf(fact_5925_acyclicI__order,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [R3: set @ ( product_prod @ B @ B ),F3: B > A] :
( ! [A5: B,B4: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A5 @ B4 ) @ R3 )
=> ( ord_less @ A @ ( F3 @ B4 ) @ ( F3 @ A5 ) ) )
=> ( transitive_acyclic @ B @ R3 ) ) ) ).
% acyclicI_order
thf(fact_5926_filtermap__nhds__times,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C3 ) @ ( topolo7230453075368039082e_nhds @ A @ A3 ) )
= ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C3 @ A3 ) ) ) ) ) ).
% filtermap_nhds_times
thf(fact_5927_acyclic__def,axiom,
! [A: $tType] :
( ( transitive_acyclic @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% acyclic_def
thf(fact_5928_acyclicI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( transitive_acyclic @ A @ R3 ) ) ).
% acyclicI
thf(fact_5929_eventually__prod__sequentially,axiom,
! [P2: ( product_prod @ nat @ nat ) > $o] :
( ( eventually @ ( product_prod @ nat @ nat ) @ P2 @ ( prod_filter @ nat @ nat @ ( at_top @ nat ) @ ( at_top @ nat ) ) )
= ( ? [N6: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ N6 @ M6 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ N6 @ N5 )
=> ( P2 @ ( product_Pair @ nat @ nat @ N5 @ M6 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_5930_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_5931_filtermap__times__pos__at__right,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [C3: A,P: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C3 ) @ ( topolo174197925503356063within @ A @ P @ ( set_ord_greaterThan @ A @ P ) ) )
= ( topolo174197925503356063within @ A @ ( times_times @ A @ C3 @ P ) @ ( set_ord_greaterThan @ A @ ( times_times @ A @ C3 @ P ) ) ) ) ) ) ).
% filtermap_times_pos_at_right
thf(fact_5932_at__to__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ A @ ( inverse_inverse @ A ) @ ( at_infinity @ A ) ) ) ) ).
% at_to_infinity
thf(fact_5933_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_5934_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_5935_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_5936_CHAR__eq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) ) ) ).
% CHAR_eq_0
thf(fact_5937_of__nat__CHAR,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_CHAR
thf(fact_5938_CHAR__eqI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C3: nat] :
( ( ( semiring_1_of_nat @ A @ C3 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ( semiring_1_of_nat @ A @ X3 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ nat @ C3 @ X3 ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C3 ) ) ) ) ).
% CHAR_eqI
thf(fact_5939_of__nat__eq__0__iff__char__dvd,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ A @ N )
= ( zero_zero @ A ) )
= ( dvd_dvd @ nat @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) @ N ) ) ) ).
% of_nat_eq_0_iff_char_dvd
thf(fact_5940_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( ( semiring_1_of_nat @ A @ N5 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_5941_CHAR__eq__posI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( ( ( semiring_1_of_nat @ A @ C3 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
=> ( ( ord_less @ nat @ X3 @ C3 )
=> ( ( semiring_1_of_nat @ A @ X3 )
!= ( zero_zero @ A ) ) ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C3 ) ) ) ) ) ).
% CHAR_eq_posI
thf(fact_5942_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ( semiring_1_of_nat @ A @ N5 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_5943_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_5944_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_5945_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_5946_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F5: filter @ A,X: B] :
( ( prod_filter @ A @ B @ F5 @ ( principal @ B @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( filtermap @ A @ ( product_prod @ A @ B )
@ ^ [A7: A] : ( product_Pair @ A @ B @ A7 @ X )
@ F5 ) ) ).
% prod_filter_principal_singleton2
thf(fact_5947_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_5948_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_5949_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_5950_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_5951_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X: A,F5: filter @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ F5 )
= ( filtermap @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ F5 ) ) ).
% prod_filter_principal_singleton
thf(fact_5952_last__list__update,axiom,
! [A: $tType,Xs2: list @ A,K2: nat,X: A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( ( K2
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs2 @ K2 @ X ) )
= X ) )
& ( ( K2
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs2 @ K2 @ X ) )
= ( last @ A @ Xs2 ) ) ) ) ) ).
% last_list_update
thf(fact_5953_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
@ ^ [R: set @ ( product_prod @ A @ A ),S6: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S6 )
& ! [A7: A,B5: A,C4: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ S6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C4 ) @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R ) ) ) ) ) ) ).
% init_seg_of_def
thf(fact_5954_last__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% last_replicate
thf(fact_5955_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_5956_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_5957_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_5958_sqr_Osimps_I3_J,axiom,
! [N: num] :
( ( sqr @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ ( plus_plus @ num @ ( sqr @ N ) @ N ) ) ) ) ).
% sqr.simps(3)
thf(fact_5959_arg__min__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F3: A > B,P2: A > $o,A3: A] :
( ( inj_on @ A @ B @ F3 @ ( collect @ A @ P2 ) )
=> ( ( P2 @ A3 )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ A3 ) @ ( F3 @ Y3 ) ) )
=> ( ( lattices_ord_arg_min @ A @ B @ F3 @ P2 )
= A3 ) ) ) ) ) ).
% arg_min_inj_eq
thf(fact_5960_arg__min__nat__le,axiom,
! [A: $tType,P2: A > $o,X: A,M2: A > nat] :
( ( P2 @ X )
=> ( ord_less_eq @ nat @ ( M2 @ ( lattices_ord_arg_min @ A @ nat @ M2 @ P2 ) ) @ ( M2 @ X ) ) ) ).
% arg_min_nat_le
thf(fact_5961_arg__min__nat__lemma,axiom,
! [A: $tType,P2: A > $o,K2: A,M2: A > nat] :
( ( P2 @ K2 )
=> ( ( P2 @ ( lattices_ord_arg_min @ A @ nat @ M2 @ P2 ) )
& ! [Y6: A] :
( ( P2 @ Y6 )
=> ( ord_less_eq @ nat @ ( M2 @ ( lattices_ord_arg_min @ A @ nat @ M2 @ P2 ) ) @ ( M2 @ Y6 ) ) ) ) ) ).
% arg_min_nat_lemma
thf(fact_5962_sqr_Osimps_I2_J,axiom,
! [N: num] :
( ( sqr @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% sqr.simps(2)
thf(fact_5963_sqr_Osimps_I1_J,axiom,
( ( sqr @ one2 )
= one2 ) ).
% sqr.simps(1)
thf(fact_5964_sqr__conv__mult,axiom,
( sqr
= ( ^ [X4: num] : ( times_times @ num @ X4 @ X4 ) ) ) ).
% sqr_conv_mult
thf(fact_5965_arg__min__equality,axiom,
! [A: $tType,C: $tType] :
( ( order @ A )
=> ! [P2: C > $o,K2: C,F3: C > A] :
( ( P2 @ K2 )
=> ( ! [X3: C] :
( ( P2 @ X3 )
=> ( ord_less_eq @ A @ ( F3 @ K2 ) @ ( F3 @ X3 ) ) )
=> ( ( F3 @ ( lattices_ord_arg_min @ C @ A @ F3 @ P2 ) )
= ( F3 @ K2 ) ) ) ) ) ).
% arg_min_equality
thf(fact_5966_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_5967_pow_Osimps_I3_J,axiom,
! [X: num,Y: num] :
( ( pow @ X @ ( bit1 @ Y ) )
= ( times_times @ num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).
% pow.simps(3)
thf(fact_5968_refl__on__singleton,axiom,
! [A: $tType,X: A] : ( refl_on @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_5969_refl__onD,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ A6 @ R3 )
=> ( ( member @ A @ A3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R3 ) ) ) ).
% refl_onD
thf(fact_5970_refl__onD1,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A6 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ A @ X @ A6 ) ) ) ).
% refl_onD1
thf(fact_5971_refl__onD2,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A6 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ A @ Y @ A6 ) ) ) ).
% refl_onD2
thf(fact_5972_refl__on__domain,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( refl_on @ A @ A6 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ A @ A3 @ A6 )
& ( member @ A @ B2 @ A6 ) ) ) ) ).
% refl_on_domain
thf(fact_5973_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one2 )
= X ) ).
% pow.simps(1)
thf(fact_5974_refl__on__def_H,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A8: set @ A,R: set @ ( product_prod @ A @ A )] :
( ! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y4: A,Z4: A] :
( ( member @ A @ Y4 @ A8 )
& ( member @ A @ Z4 @ A8 ) )
@ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R ) ) ) ) ) ).
% refl_on_def'
thf(fact_5975_pow_Osimps_I2_J,axiom,
! [X: num,Y: num] :
( ( pow @ X @ ( bit0 @ Y ) )
= ( sqr @ ( pow @ X @ Y ) ) ) ).
% pow.simps(2)
thf(fact_5976_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) )
=> ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( A3 = B2 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_5977_Ex__inj__on__UNION__Sigma,axiom,
! [A: $tType,B: $tType,A6: B > ( set @ A ),I5: set @ B] :
? [F2: A > ( product_prod @ B @ A )] :
( ( inj_on @ A @ ( product_prod @ B @ A ) @ F2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I5 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ A @ ( product_prod @ B @ A ) @ F2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I5 ) ) ) @ ( product_Sigma @ B @ A @ I5 @ A6 ) ) ) ).
% Ex_inj_on_UNION_Sigma
thf(fact_5978_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
= ( ( member @ A @ A3 @ A6 )
& ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_5979_SigmaI,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B2: B,B6: A > ( set @ B )] :
( ( member @ A @ A3 @ A6 )
=> ( ( member @ B @ B2 @ ( B6 @ A3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) ) ) ) ).
% SigmaI
thf(fact_5980_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P2: A > $o,Q: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] :
( ( P2 @ A7 )
& ( Q @ B5 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P2 )
@ ^ [Uu3: A] : ( collect @ B @ Q ) ) ) ).
% Collect_case_prod
thf(fact_5981_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_5982_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_5983_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_5984_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_5985_Times__empty,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
( ( ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( B6
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_5986_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_5987_fst__image__times,axiom,
! [B: $tType,A: $tType,B6: set @ B,A6: set @ A] :
( ( ( B6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 ) )
= A6 ) ) ) ).
% fst_image_times
thf(fact_5988_snd__image__times,axiom,
! [B: $tType,A: $tType,A6: set @ B,B6: set @ A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu3: B] : B6 ) )
= ( 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] : B6 ) )
= B6 ) ) ) ).
% snd_image_times
thf(fact_5989_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B2: B,B6: set @ B] :
( ( product_Sigma @ A @ B @ ( insert @ A @ A3 @ A6 )
@ ^ [Uu3: A] : ( insert @ B @ B2 @ B6 ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : ( insert @ B @ B2 @ B6 ) )
@ ( product_Sigma @ A @ B @ ( insert @ A @ A3 @ A6 )
@ ^ [Uu3: A] : B6 ) ) ) ) ).
% insert_Times_insert
thf(fact_5990_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_5991_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_5992_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P2: A > $o,Q: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y4: B] :
( ( P2 @ X4 )
& ( Q @ X4 @ Y4 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P2 )
@ ^ [X4: A] : ( collect @ B @ ( Q @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_5993_Sigma__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,C5: set @ A,B6: A > ( set @ B ),D5: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ C5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( B6 @ X3 ) @ ( D5 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) @ ( product_Sigma @ A @ B @ C5 @ D5 ) ) ) ) ).
% Sigma_mono
thf(fact_5994_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J4: set @ A,C5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ I5 @ J4 ) @ C5 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% Sigma_Int_distrib1
thf(fact_5995_Sigma__Diff__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J4: set @ A,C5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I5 @ J4 ) @ C5 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% Sigma_Diff_distrib1
thf(fact_5996_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J4: set @ A,C5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I5 @ J4 ) @ C5 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_5997_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C5: set @ A,A6: set @ B,B6: set @ B] :
( ( member @ A @ X @ C5 )
=> ( ( ( product_Sigma @ B @ A @ A6
@ ^ [Uu3: B] : C5 )
= ( product_Sigma @ B @ A @ B6
@ ^ [Uu3: B] : C5 ) )
= ( A6 = B6 ) ) ) ).
% Times_eq_cancel2
thf(fact_5998_Sigma__cong,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,C5: A > ( set @ B ),D5: A > ( set @ B )] :
( ( A6 = B6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B6 )
=> ( ( C5 @ X3 )
= ( D5 @ X3 ) ) )
=> ( ( product_Sigma @ A @ B @ A6 @ C5 )
= ( product_Sigma @ A @ B @ B6 @ D5 ) ) ) ) ).
% Sigma_cong
thf(fact_5999_times__eq__iff,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B,C5: set @ A,D5: set @ B] :
( ( ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 )
= ( product_Sigma @ A @ B @ C5
@ ^ [Uu3: A] : D5 ) )
= ( ( ( A6 = C5 )
& ( B6 = D5 ) )
| ( ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( B6
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C5
= ( bot_bot @ ( set @ A ) ) )
| ( D5
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_6000_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A8: set @ A,B8: set @ B] :
( product_Sigma @ A @ B @ A8
@ ^ [Uu3: A] : B8 ) ) ) ).
% Product_Type.product_def
thf(fact_6001_member__product,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A6 @ B6 ) )
= ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 ) ) ) ).
% member_product
thf(fact_6002_wo__rel_Omax2__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 )
= B2 ) )
& ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 )
= A3 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_6003_wo__rel_OTOTALS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ! [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R3 ) )
=> ! [Xa: A] :
( ( member @ A @ Xa @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa ) @ R3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X5 ) @ R3 ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_6004_SigmaE2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
=> ~ ( ( member @ A @ A3 @ A6 )
=> ~ ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_6005_SigmaD2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
=> ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ).
% SigmaD2
thf(fact_6006_SigmaD1,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
=> ( member @ A @ A3 @ A6 ) ) ).
% SigmaD1
thf(fact_6007_SigmaE,axiom,
! [A: $tType,B: $tType,C3: product_prod @ A @ B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ! [Y3: B] :
( ( member @ B @ Y3 @ ( B6 @ X3 ) )
=> ( C3
!= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_6008_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P2: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ! [X3: A] :
( ! [Y6: A] :
( ( ( Y6 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R3 ) )
=> ( P2 @ Y6 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A3 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_6009_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C5: set @ A,A6: set @ B,B6: set @ B] :
( ( member @ A @ X @ C5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu3: B] : C5 )
@ ( product_Sigma @ B @ A @ B6
@ ^ [Uu3: B] : C5 ) )
= ( ord_less_eq @ ( set @ B ) @ A6 @ B6 ) ) ) ).
% Times_subset_cancel2
thf(fact_6010_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A6: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A6 )
& ( member @ B @ ( product_snd @ A @ B @ X ) @ B6 ) ) ) ).
% mem_Times_iff
thf(fact_6011_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I5: set @ A,X8: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I5 @ X8 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ I5 )
=> ( ( X8 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_6012_Times__Int__distrib1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A6 @ B6 )
@ ^ [Uu3: A] : C5 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu3: A] : C5 ) ) ) ).
% Times_Int_distrib1
thf(fact_6013_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A6: A > ( set @ B ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I: A] : ( inf_inf @ ( set @ B ) @ ( A6 @ I ) @ ( B6 @ I ) ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A6 ) @ ( product_Sigma @ A @ B @ I5 @ B6 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_6014_Times__Int__Times,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B,C5: set @ A,D5: set @ B] :
( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 )
@ ( product_Sigma @ A @ B @ C5
@ ^ [Uu3: A] : D5 ) )
= ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A6 @ C5 )
@ ^ [Uu3: A] : ( inf_inf @ ( set @ B ) @ B6 @ D5 ) ) ) ).
% Times_Int_Times
thf(fact_6015_infinite__cartesian__product,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ~ ( finite_finite2 @ B @ B6 )
=> ~ ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_6016_wo__rel_Omax2__equals1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 )
= A3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_6017_wo__rel_Omax2__equals2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 )
= B2 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_6018_wo__rel_Omax2__greater,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_6019_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A6: A > ( set @ B ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I: A] : ( sup_sup @ ( set @ B ) @ ( A6 @ I ) @ ( B6 @ I ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A6 ) @ ( product_Sigma @ A @ B @ I5 @ B6 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_6020_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A6 @ B6 )
@ ^ [Uu3: A] : C5 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu3: A] : C5 ) ) ) ).
% Times_Un_distrib1
thf(fact_6021_Sigma__Union,axiom,
! [B: $tType,A: $tType,X8: set @ ( set @ A ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X8 ) @ B6 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A8: set @ A] : ( product_Sigma @ A @ B @ A8 @ B6 )
@ X8 ) ) ) ).
% Sigma_Union
thf(fact_6022_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A6: A > ( set @ B ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I: A] : ( minus_minus @ ( set @ B ) @ ( A6 @ I ) @ ( B6 @ I ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A6 ) @ ( product_Sigma @ A @ B @ I5 @ B6 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_6023_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A6 @ B6 )
@ ^ [Uu3: A] : C5 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu3: A] : C5 ) ) ) ).
% Times_Diff_distrib1
thf(fact_6024_times__subset__iff,axiom,
! [A: $tType,B: $tType,A6: set @ A,C5: set @ B,B6: set @ A,D5: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu3: A] : D5 ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( C5
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ B ) @ C5 @ D5 ) ) ) ) ).
% times_subset_iff
thf(fact_6025_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) )
=> ( ( A3 = B2 )
| ( member @ A @ A3 @ A6 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_6026_wfI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : B6 ) )
=> ( ! [X3: A,P8: A > $o] :
( ! [Xa: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Xa ) @ R3 )
=> ( P8 @ Y3 ) )
=> ( P8 @ Xa ) )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( member @ A @ X3 @ B6 )
=> ( P8 @ X3 ) ) ) )
=> ( wf @ A @ R3 ) ) ) ).
% wfI
thf(fact_6027_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
= ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ( B6 @ X4 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_6028_refl__onI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu3: A] : A6 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 ) )
=> ( refl_on @ A @ A6 @ R3 ) ) ) ).
% refl_onI
thf(fact_6029_refl__on__def,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A8: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A8
@ ^ [Uu3: A] : A8 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R ) ) ) ) ) ).
% refl_on_def
thf(fact_6030_UN__Times__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,E5: C > ( set @ A ),F5: D > ( set @ B ),A6: set @ C,B6: set @ D] :
( ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ C @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A7: C,B5: D] :
( product_Sigma @ A @ B @ ( E5 @ A7 )
@ ^ [Uu3: A] : ( F5 @ B5 ) ) )
@ ( product_Sigma @ C @ D @ A6
@ ^ [Uu3: C] : B6 ) ) )
= ( 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 ) @ F5 @ B6 ) ) ) ) ).
% UN_Times_distrib
thf(fact_6031_swap__product,axiom,
! [B: $tType,A: $tType,A6: set @ B,B6: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I: B,J: A] : ( product_Pair @ A @ B @ J @ I ) )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu3: B] : B6 ) )
= ( product_Sigma @ A @ B @ B6
@ ^ [Uu3: A] : A6 ) ) ).
% swap_product
thf(fact_6032_card__cartesian__product,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 ) )
= ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B6 ) ) ) ).
% card_cartesian_product
thf(fact_6033_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,B6: set @ D,B13: set @ C] :
( ( ( image2 @ B @ A @ F3 @ A6 )
= A11 )
=> ( ( ( image2 @ D @ C @ G3 @ B6 )
= B13 )
=> ( ( image2 @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F3 @ G3 )
@ ( product_Sigma @ B @ D @ A6
@ ^ [Uu3: B] : B6 ) )
= ( product_Sigma @ A @ C @ A11
@ ^ [Uu3: A] : B13 ) ) ) ) ).
% map_prod_surj_on
thf(fact_6034_map__prod__inj__on,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F3: A > B,A6: set @ A,G3: C > D,B6: set @ C] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( inj_on @ C @ D @ G3 @ B6 )
=> ( inj_on @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G3 )
@ ( product_Sigma @ A @ C @ A6
@ ^ [Uu3: A] : B6 ) ) ) ) ).
% map_prod_inj_on
thf(fact_6035_wo__rel_Ocases__Total,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_6036_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,Y4: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y4 @ N )
& ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_6037_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 )
& ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_6038_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A6: set @ B,B6: B > ( set @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A6 @ B6 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% snd_image_Sigma
thf(fact_6039_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_6040_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F3: C > A,G3: D > B,A6: set @ C,B6: set @ D] :
( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X4: C,Y4: D] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ Y4 ) ) )
@ ( product_Sigma @ C @ D @ A6
@ ^ [Uu3: C] : B6 ) )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F3 @ A6 )
@ ^ [Uu3: A] : ( image2 @ D @ B @ G3 @ B6 ) ) ) ).
% image_paired_Times
thf(fact_6041_uniformly__continuous__on__uniformity,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo7287701948861334536_space @ A )
& ( topolo7287701948861334536_space @ B ) )
=> ( ( topolo6026614971017936543ous_on @ A @ B )
= ( ^ [S6: 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,Y4: A] : ( product_Pair @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) )
@ ( topolo7806501430040627800ormity @ B )
@ ( inf_inf @ ( filter @ ( product_prod @ A @ A ) ) @ ( topolo7806501430040627800ormity @ A )
@ ( principal @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ S6
@ ^ [Uu3: A] : S6 ) ) ) ) ) ) ) ).
% uniformly_continuous_on_uniformity
thf(fact_6042_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,N5: A] : ( cons @ A @ N5 @ 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_6043_pairs__le__eq__Sigma,axiom,
! [M2: nat] :
( ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I: nat,J: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ J ) @ M2 ) ) )
= ( product_Sigma @ nat @ nat @ ( set_ord_atMost @ nat @ M2 )
@ ^ [R: nat] : ( set_ord_atMost @ nat @ ( minus_minus @ nat @ M2 @ R ) ) ) ) ).
% pairs_le_eq_Sigma
thf(fact_6044_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A8: set @ A,B8: 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 ) )
@ ^ [Y4: B] : ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B8 @ X4 ) ) )
@ A8 ) ) ) ) ).
% Sigma_def
thf(fact_6045_product__fold,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: A,Z4: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y4: B] : ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) )
@ Z4
@ B6 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A6 ) ) ) ) ).
% product_fold
thf(fact_6046_well__order__induct__imp,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P2: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ! [X3: A] :
( ! [Y6: A] :
( ( ( Y6 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R3 ) )
=> ( ( member @ A @ Y6 @ ( field2 @ A @ R3 ) )
=> ( P2 @ Y6 ) ) )
=> ( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
=> ( P2 @ X3 ) ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( P2 @ A3 ) ) ) ) ).
% well_order_induct_imp
thf(fact_6047_wo__rel_Ominim__least,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B6: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ B6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B6 ) @ B2 ) @ R3 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_6048_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R3: A,S2: B,R2: set @ ( product_prod @ A @ B ),S8: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S2 ) @ R2 )
=> ( ( S8 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S8 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_6049_wo__rel_Oequals__minim,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B6: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ A3 @ B6 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B4 ) @ R3 ) )
=> ( A3
= ( bNF_We6954850376910717587_minim @ A @ R3 @ B6 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_6050_mult__inj__if__coprime__nat,axiom,
! [B: $tType,A: $tType,F3: A > nat,A6: set @ A,G3: B > nat,B6: set @ B] :
( ( inj_on @ A @ nat @ F3 @ A6 )
=> ( ( inj_on @ B @ nat @ G3 @ B6 )
=> ( ! [A5: A,B4: B] :
( ( member @ A @ A5 @ A6 )
=> ( ( member @ B @ B4 @ B6 )
=> ( algebr8660921524188924756oprime @ nat @ ( F3 @ A5 ) @ ( G3 @ B4 ) ) ) )
=> ( inj_on @ ( product_prod @ A @ B ) @ nat
@ ( product_case_prod @ A @ B @ nat
@ ^ [A7: A,B5: B] : ( times_times @ nat @ ( F3 @ A7 ) @ ( G3 @ B5 ) ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 ) ) ) ) ) ).
% mult_inj_if_coprime_nat
thf(fact_6051_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 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_6052_coprime__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ C3 @ ( times_times @ A @ A3 @ B2 ) )
= ( ( algebr8660921524188924756oprime @ A @ C3 @ A3 )
& ( algebr8660921524188924756oprime @ A @ C3 @ B2 ) ) ) ) ).
% coprime_mult_right_iff
thf(fact_6053_coprime__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ C3 )
& ( algebr8660921524188924756oprime @ A @ B2 @ C3 ) ) ) ) ).
% coprime_mult_left_iff
thf(fact_6054_coprime__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_self
thf(fact_6055_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_6056_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_6057_coprime__mod__right__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ ( modulo_modulo @ A @ B2 @ A3 ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mod_right_iff
thf(fact_6058_coprime__mod__left__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mod_left_iff
thf(fact_6059_coprime__0__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( zero_zero @ A ) )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_right_iff
thf(fact_6060_coprime__0__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_left_iff
thf(fact_6061_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_left_iff
thf(fact_6062_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_right_iff
thf(fact_6063_gcd__mult__left__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( algebr8660921524188924756oprime @ A @ B2 @ C3 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ C3 @ A3 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_left_left_cancel
thf(fact_6064_gcd__mult__left__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( algebr8660921524188924756oprime @ A @ B2 @ C3 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_left_right_cancel
thf(fact_6065_gcd__mult__right__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C3 )
=> ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_right_left_cancel
thf(fact_6066_gcd__mult__right__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C3 )
=> ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_right_right_cancel
thf(fact_6067_finite__vimage__Suc__iff,axiom,
! [F5: set @ nat] :
( ( finite_finite2 @ nat @ ( vimage @ nat @ nat @ suc @ F5 ) )
= ( finite_finite2 @ nat @ F5 ) ) ).
% finite_vimage_Suc_iff
thf(fact_6068_coprime__Suc__0__right,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ).
% coprime_Suc_0_right
thf(fact_6069_coprime__Suc__0__left,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ).
% coprime_Suc_0_left
thf(fact_6070_coprime__divisors,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A,B2: A,D3: A] :
( ( dvd_dvd @ A @ A3 @ C3 )
=> ( ( dvd_dvd @ A @ B2 @ D3 )
=> ( ( algebr8660921524188924756oprime @ A @ C3 @ D3 )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% coprime_divisors
thf(fact_6071_coprime__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [B5: A,A7: A] : ( algebr8660921524188924756oprime @ A @ A7 @ B5 ) ) ) ) ).
% coprime_commute
thf(fact_6072_coprime__Suc__left__nat,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ N ) @ N ) ).
% coprime_Suc_left_nat
thf(fact_6073_coprime__Suc__right__nat,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ N ) ) ).
% coprime_Suc_right_nat
thf(fact_6074_coprime__1__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( one_one @ A ) ) ) ).
% coprime_1_right
thf(fact_6075_coprime__1__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A3 ) ) ).
% coprime_1_left
thf(fact_6076_coprime__crossproduct__nat,axiom,
! [A3: nat,D3: nat,B2: nat,C3: nat] :
( ( algebr8660921524188924756oprime @ nat @ A3 @ D3 )
=> ( ( algebr8660921524188924756oprime @ nat @ B2 @ C3 )
=> ( ( ( times_times @ nat @ A3 @ C3 )
= ( times_times @ nat @ B2 @ D3 ) )
= ( ( A3 = B2 )
& ( C3 = D3 ) ) ) ) ) ).
% coprime_crossproduct_nat
thf(fact_6077_mult__mod__cancel__right,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [A3: A,N: A,M2: A,B2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ N ) @ M2 )
= ( modulo_modulo @ A @ ( times_times @ A @ B2 @ N ) @ M2 ) )
=> ( ( algebr8660921524188924756oprime @ A @ M2 @ N )
=> ( ( modulo_modulo @ A @ A3 @ M2 )
= ( modulo_modulo @ A @ B2 @ M2 ) ) ) ) ) ).
% mult_mod_cancel_right
thf(fact_6078_mult__mod__cancel__left,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [N: A,A3: A,M2: A,B2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ N @ A3 ) @ M2 )
= ( modulo_modulo @ A @ ( times_times @ A @ N @ B2 ) @ M2 ) )
=> ( ( algebr8660921524188924756oprime @ A @ M2 @ N )
=> ( ( modulo_modulo @ A @ A3 @ M2 )
= ( modulo_modulo @ A @ B2 @ M2 ) ) ) ) ) ).
% mult_mod_cancel_left
thf(fact_6079_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_right_imp_coprime
thf(fact_6080_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_left_imp_coprime
thf(fact_6081_coprime__common__divisor,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) ) ) ) ).
% coprime_common_divisor
thf(fact_6082_coprime__absorb__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [Y: A,X: A] :
( ( dvd_dvd @ A @ Y @ X )
=> ( ( algebr8660921524188924756oprime @ A @ X @ Y )
= ( dvd_dvd @ A @ Y @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_right
thf(fact_6083_coprime__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,D3: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ C3 @ D3 )
=> ( ! [E2: A] :
( ~ ( dvd_dvd @ A @ E2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E2 @ A3 )
=> ( ( dvd_dvd @ A @ E2 @ B2 )
=> ( dvd_dvd @ A @ E2 @ C3 ) ) ) )
=> ( ! [E2: A] :
( ~ ( dvd_dvd @ A @ E2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E2 @ A3 )
=> ( ( dvd_dvd @ A @ E2 @ B2 )
=> ( dvd_dvd @ A @ E2 @ D3 ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% coprime_imp_coprime
thf(fact_6084_coprime__absorb__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( algebr8660921524188924756oprime @ A @ X @ Y )
= ( dvd_dvd @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_left
thf(fact_6085_not__coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ~ ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% not_coprimeI
thf(fact_6086_not__coprimeE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ~ ! [C2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) ) ) ) ).
% not_coprimeE
thf(fact_6087_coprime__def,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A7: A,B5: A] :
! [C4: A] :
( ( dvd_dvd @ A @ C4 @ A7 )
=> ( ( dvd_dvd @ A @ C4 @ B5 )
=> ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) ) ) ) ) ).
% coprime_def
thf(fact_6088_coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ! [C2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprimeI
thf(fact_6089_coprime__dvd__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C3 )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% coprime_dvd_mult_right_iff
thf(fact_6090_coprime__dvd__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C3 )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% coprime_dvd_mult_left_iff
thf(fact_6091_divides__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C3 )
=> ( ( dvd_dvd @ A @ B2 @ C3 )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ) ) ).
% divides_mult
thf(fact_6092_vimage__Suc__insert__Suc,axiom,
! [N: nat,A6: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( suc @ N ) @ A6 ) )
= ( insert @ nat @ N @ ( vimage @ nat @ nat @ suc @ A6 ) ) ) ).
% vimage_Suc_insert_Suc
thf(fact_6093_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > ( product_prod @ B @ C ),A6: set @ B,B6: set @ C] :
( ( vimage @ A @ ( product_prod @ B @ C ) @ F3
@ ( product_Sigma @ B @ C @ A6
@ ^ [Uu3: B] : B6 ) )
= ( 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 ) @ B6 ) ) ) ).
% vimage_Times
thf(fact_6094_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,A6: set @ B,F3: B > ( set @ A )] :
( ( ( member @ B @ X @ A6 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A6 @ F3 ) )
= ( F3 @ X ) ) )
& ( ~ ( member @ B @ X @ A6 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A6 @ F3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_6095_vimage__Suc__insert__0,axiom,
! [A6: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( zero_zero @ nat ) @ A6 ) )
= ( vimage @ nat @ nat @ suc @ A6 ) ) ).
% vimage_Suc_insert_0
thf(fact_6096_invertible__coprime,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ C3 ) ) ) ).
% invertible_coprime
thf(fact_6097_gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,A4: A,B3: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
= ( times_times @ A @ A4 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( ( B2
= ( times_times @ A @ B3 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ) ).
% gcd_coprime
thf(fact_6098_gcd__coprime__exists,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ? [A17: A,B10: A] :
( ( A3
= ( times_times @ A @ A17 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( B2
= ( times_times @ A @ B10 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( algebr8660921524188924756oprime @ A @ A17 @ B10 ) ) ) ) ).
% gcd_coprime_exists
thf(fact_6099_div__gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( A3
!= ( zero_zero @ A ) )
| ( B2
!= ( zero_zero @ A ) ) )
=> ( algebr8660921524188924756oprime @ A @ ( divide_divide @ A @ A3 @ ( gcd_gcd @ A @ A3 @ B2 ) ) @ ( divide_divide @ A @ B2 @ ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ) ).
% div_gcd_coprime
thf(fact_6100_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_6101_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_6102_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ N ) ) ).
% coprime_diff_one_left_nat
thf(fact_6103_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_6104_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F3: A > B,D5: set @ A,A6: set @ B] :
( ( inj_on @ A @ B @ F3 @ D5 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A6 ) @ D5 ) ) @ ( finite_card @ B @ A6 ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_6105_Rats__abs__nat__div__natE,axiom,
! [X: real] :
( ( member @ real @ X @ ( field_char_0_Rats @ real ) )
=> ~ ! [M: nat,N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
=> ( ( ( abs_abs @ real @ X )
= ( divide_divide @ real @ ( semiring_1_of_nat @ real @ M ) @ ( semiring_1_of_nat @ real @ N3 ) ) )
=> ~ ( algebr8660921524188924756oprime @ nat @ M @ N3 ) ) ) ) ).
% Rats_abs_nat_div_natE
thf(fact_6106_set__decode__div__2,axiom,
! [X: nat] :
( ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( vimage @ nat @ nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% set_decode_div_2
thf(fact_6107_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_6108_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,Y4: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y4 @ N )
& ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_6109_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_0 @ B )
=> ! [F3: A > B,G3: C > B,A6: set @ A,B6: set @ C] :
( ( inj_on @ ( product_prod @ A @ C ) @ B
@ ( product_case_prod @ A @ C @ B
@ ^ [A7: A,B5: C] : ( times_times @ B @ ( F3 @ A7 ) @ ( G3 @ B5 ) ) )
@ ( product_Sigma @ A @ C @ A6
@ ^ [Uu3: A] : B6 ) )
=> ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A6 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G3 @ B6 ) )
= ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B )
@ ( collect @ B
@ ^ [Uu3: B] :
? [A7: A,B5: C] :
( ( Uu3
= ( times_times @ B @ ( F3 @ A7 ) @ ( G3 @ B5 ) ) )
& ( member @ A @ A7 @ A6 )
& ( member @ C @ B5 @ B6 ) ) ) ) ) ) ) ).
% sum_mult_sum_if_inj
thf(fact_6110_of__nat__eq__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( id @ nat ) ) ).
% of_nat_eq_id
thf(fact_6111_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_6112_id__funpow,axiom,
! [A: $tType,N: nat] :
( ( compow @ ( A > A ) @ N @ ( id @ A ) )
= ( id @ A ) ) ).
% id_funpow
thf(fact_6113_rotate0,axiom,
! [A: $tType] :
( ( rotate @ A @ ( zero_zero @ nat ) )
= ( id @ ( list @ A ) ) ) ).
% rotate0
thf(fact_6114_apfst__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apfst @ A @ A @ B @ ( id @ A ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apfst_id
thf(fact_6115_apsnd__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apsnd_id
thf(fact_6116_push__bit__0__id,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% push_bit_0_id
thf(fact_6117_comp__the__Some,axiom,
! [A: $tType] :
( ( comp @ ( option @ A ) @ A @ A @ ( the2 @ A ) @ ( some @ A ) )
= ( id @ A ) ) ).
% comp_the_Some
thf(fact_6118_drop__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% drop_bit_0
thf(fact_6119_coprime__crossproduct__int,axiom,
! [A3: int,D3: int,B2: int,C3: int] :
( ( algebr8660921524188924756oprime @ int @ A3 @ D3 )
=> ( ( algebr8660921524188924756oprime @ int @ B2 @ C3 )
=> ( ( ( times_times @ int @ ( abs_abs @ int @ A3 ) @ ( abs_abs @ int @ C3 ) )
= ( times_times @ int @ ( abs_abs @ int @ B2 ) @ ( abs_abs @ int @ D3 ) ) )
= ( ( ( abs_abs @ int @ A3 )
= ( abs_abs @ int @ B2 ) )
& ( ( abs_abs @ int @ C3 )
= ( abs_abs @ int @ D3 ) ) ) ) ) ) ).
% coprime_crossproduct_int
thf(fact_6120_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_6121_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_6122_natLeq__Linear__order,axiom,
order_679001287576687338der_on @ nat @ ( field2 @ nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).
% natLeq_Linear_order
thf(fact_6123_natLeq__Total,axiom,
total_on @ nat @ ( field2 @ nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).
% natLeq_Total
thf(fact_6124_natLeq__Refl,axiom,
refl_on @ nat @ ( field2 @ nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).
% natLeq_Refl
thf(fact_6125_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_6126_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_6127_natLeq__natLess__Id,axiom,
( bNF_Ca8459412986667044542atLess
= ( minus_minus @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq @ ( id2 @ nat ) ) ) ).
% natLeq_natLess_Id
thf(fact_6128_Field__natLeq,axiom,
( ( field2 @ nat @ bNF_Ca8665028551170535155natLeq )
= ( top_top @ ( set @ nat ) ) ) ).
% Field_natLeq
thf(fact_6129_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% natLeq_def
thf(fact_6130_fst__diag__id,axiom,
! [A: $tType,Z3: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Z3 )
= ( id @ A @ Z3 ) ) ).
% fst_diag_id
thf(fact_6131_snd__diag__id,axiom,
! [A: $tType,Z3: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Z3 )
= ( id @ A @ Z3 ) ) ).
% snd_diag_id
thf(fact_6132_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,Y4: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y4 @ N )
& ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_6133_relation__of__def,axiom,
! [A: $tType] :
( ( order_relation_of @ A )
= ( ^ [P4: A > A > $o,A8: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A7: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 )
@ ( product_Sigma @ A @ A @ A8
@ ^ [Uu3: A] : A8 ) )
& ( P4 @ A7 @ B5 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_6134_underS__I,axiom,
! [A: $tType,I2: A,J2: A,R2: set @ ( product_prod @ A @ A )] :
( ( I2 != J2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J2 ) @ R2 )
=> ( member @ A @ I2 @ ( order_underS @ A @ R2 @ J2 ) ) ) ) ).
% underS_I
thf(fact_6135_underS__E,axiom,
! [A: $tType,I2: A,R2: set @ ( product_prod @ A @ A ),J2: A] :
( ( member @ A @ I2 @ ( order_underS @ A @ R2 @ J2 ) )
=> ( ( I2 != J2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J2 ) @ R2 ) ) ) ).
% underS_E
thf(fact_6136_underS__def,axiom,
! [A: $tType] :
( ( order_underS @ A )
= ( ^ [R: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B5: A] :
( ( B5 != A7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R ) ) ) ) ) ).
% underS_def
thf(fact_6137_natLeq__underS__less,axiom,
! [N: nat] :
( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
= ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N ) ) ) ).
% natLeq_underS_less
thf(fact_6138_underS__incl__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( order_underS @ A @ R3 @ B2 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_6139_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_6140_sorted__insort__insert__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs2: list @ B,X: 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 @ X @ Xs2 ) ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_6141_swap__swap,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
= P ) ).
% swap_swap
thf(fact_6142_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= ( product_Pair @ A @ B @ Y @ X ) ) ).
% swap_simp
thf(fact_6143_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B > A,P: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y4: B,X4: C] : ( F3 @ X4 @ Y4 )
@ ( product_swap @ C @ B @ P ) )
= ( product_case_prod @ C @ B @ A @ F3 @ P ) ) ).
% case_swap
thf(fact_6144_fst__swap,axiom,
! [A: $tType,B: $tType,X: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
= ( product_snd @ B @ A @ X ) ) ).
% fst_swap
thf(fact_6145_snd__swap,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
= ( product_fst @ A @ B @ X ) ) ).
% snd_swap
thf(fact_6146_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,A6: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A6 ) ) ).
% pair_in_swap_image
thf(fact_6147_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_6148_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_6149_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_6150_product__swap,axiom,
! [B: $tType,A: $tType,A6: set @ B,B6: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu3: B] : B6 ) )
= ( product_Sigma @ A @ B @ B6
@ ^ [Uu3: A] : A6 ) ) ).
% product_swap
thf(fact_6151_sorted__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,X: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs2 ) ) ) ) ).
% sorted_insort_insert
thf(fact_6152_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_6153_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_6154_pair__lessI2,axiom,
! [A3: nat,B2: nat,S2: nat,T2: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less @ nat @ S2 @ 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 @ S2 ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_6155_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ X @ Z3 ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y @ Z3 ) ) ).
% pair_less_iff1
thf(fact_6156_VEBT_Osimps_I8_J,axiom,
! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A,F22: $o > $o > A,X21: $o,X22: $o] :
( ( vEBT_rec_VEBT @ A @ F1 @ F22 @ ( vEBT_Leaf @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% VEBT.simps(8)
thf(fact_6157_pair__lessI1,axiom,
! [A3: nat,B2: nat,S2: 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 @ S2 ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_6158_pair__leqI2,axiom,
! [A3: nat,B2: nat,S2: nat,T2: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less_eq @ nat @ S2 @ 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 @ S2 ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_6159_pair__leqI1,axiom,
! [A3: nat,B2: nat,S2: 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 @ S2 ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_6160_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_6161_length__upto,axiom,
! [I2: int,J2: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I2 @ J2 ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J2 @ I2 ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_6162_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_6163_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_6164_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_6165_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_6166_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_6167_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top @ nat @ ( dvd_dvd @ nat )
@ ^ [M6: nat,N5: nat] :
( ( dvd_dvd @ nat @ M6 @ N5 )
& ( M6 != N5 ) )
@ ( zero_zero @ nat ) ) ).
% gcd_nat.ordering_top_axioms
thf(fact_6168_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_6169_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top @ nat
@ ^ [X4: nat,Y4: nat] : ( ord_less_eq @ nat @ Y4 @ X4 )
@ ^ [X4: nat,Y4: nat] : ( ord_less @ nat @ Y4 @ X4 )
@ ( zero_zero @ nat ) ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_6170_ratrel__iff,axiom,
( ratrel
= ( ^ [X4: product_prod @ int @ int,Y4: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X4 )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y4 )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X4 ) ) ) ) ) ) ).
% ratrel_iff
thf(fact_6171_dropWhile__nth,axiom,
! [A: $tType,J2: nat,P2: A > $o,Xs2: list @ A] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P2 @ Xs2 ) ) )
=> ( ( nth @ A @ ( dropWhile @ A @ P2 @ Xs2 ) @ J2 )
= ( nth @ A @ Xs2 @ ( plus_plus @ nat @ J2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs2 ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_6172_length__dropWhile__le,axiom,
! [A: $tType,P2: A > $o,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P2 @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_dropWhile_le
thf(fact_6173_sorted__dropWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,P2: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( dropWhile @ A @ P2 @ Xs2 ) ) ) ) ).
% sorted_dropWhile
thf(fact_6174_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_6175_ratrel__def,axiom,
( ratrel
= ( ^ [X4: product_prod @ int @ int,Y4: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X4 )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y4 )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X4 ) ) ) ) ) ) ).
% ratrel_def
thf(fact_6176_plus__rat_Oabs__eq,axiom,
! [Xa2: product_prod @ int @ int,X: product_prod @ int @ int] :
( ( ratrel @ Xa2 @ Xa2 )
=> ( ( ratrel @ X @ X )
=> ( ( plus_plus @ rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa2 ) @ ( product_snd @ int @ int @ X ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa2 ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_6177_times__rat_Oabs__eq,axiom,
! [Xa2: product_prod @ int @ int,X: product_prod @ int @ int] :
( ( ratrel @ Xa2 @ Xa2 )
=> ( ( ratrel @ X @ X )
=> ( ( times_times @ rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa2 ) @ ( product_fst @ int @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa2 ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% times_rat.abs_eq
thf(fact_6178_Rat_Opositive_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( positive @ ( abs_Rat @ X ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% Rat.positive.abs_eq
thf(fact_6179_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P4: A > $o,Xs: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X4: A,Xa3: list @ A] : ( some @ A @ X4 )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs ) ) ) ) ).
% find_dropWhile
thf(fact_6180_Rat_Opositive__mult,axiom,
! [X: rat,Y: rat] :
( ( positive @ X )
=> ( ( positive @ Y )
=> ( positive @ ( times_times @ rat @ X @ Y ) ) ) ) ).
% Rat.positive_mult
thf(fact_6181_Rat_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X4: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ ( rep_Rat @ X4 ) ) @ ( product_snd @ int @ int @ ( rep_Rat @ X4 ) ) ) ) ) ) ).
% Rat.positive.rep_eq
thf(fact_6182_Rat_Opositive_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o @ ratrel
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ ^ [X4: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ X4 ) ) )
@ ^ [X4: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ X4 ) ) ) ) ).
% Rat.positive.rsp
thf(fact_6183_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( ring_1 @ B )
& ( ring_1 @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
=> ( bNF_rel_fun @ int @ int @ A @ B
@ ^ [Y5: int,Z2: int] : Y5 = Z2
@ R2
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_6184_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( semiring_numeral @ B )
& ( monoid_add @ A )
& ( semiring_numeral @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y5: num,Z2: num] : Y5 = Z2
@ R2
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_6185_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( power @ B )
& ( power @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R2
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ R2 )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_6186_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( semiring_1 @ B )
& ( semiring_1 @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ R2
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_6187_Rat_Opositive__def,axiom,
( positive
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o )
@ ^ [X4: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ X4 ) ) ) ) ) ).
% Rat.positive_def
thf(fact_6188_less__eq__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( ord_less_eq @ nat )
@ ( ord_less_eq @ nat ) ) ).
% less_eq_natural.rsp
thf(fact_6189_Suc_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ suc
@ suc ) ).
% Suc.rsp
thf(fact_6190_times__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y5: int,Z2: int] : Y5 = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z2: int] : Y5 = Z2
@ ^ [Y5: int,Z2: int] : Y5 = Z2 )
@ ( times_times @ int )
@ ( times_times @ int ) ) ).
% times_integer.rsp
thf(fact_6191_times__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
@ ( times_times @ nat )
@ ( times_times @ nat ) ) ).
% times_natural.rsp
thf(fact_6192_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( ( zero_neq_one @ B )
& ( zero_neq_one @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ R2
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_6193_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,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
@ ^ [X4: product_prod @ int @ int,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) ) ) ).
% times_rat.rsp
thf(fact_6194_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,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
@ ^ [X4: product_prod @ int @ int,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) ) ) ).
% plus_rat.rsp
thf(fact_6195_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,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
@ ( plus_plus @ rat ) ) ).
% plus_rat.transfer
thf(fact_6196_Rat_Opositive_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ $o @ $o @ pcr_rat
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ ^ [X4: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ X4 ) ) )
@ positive ) ).
% Rat.positive.transfer
thf(fact_6197_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,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
@ ( times_times @ rat ) ) ).
% times_rat.transfer
thf(fact_6198_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y4 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V4 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_6199_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_6200_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_6201_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ pcr_int
@ ^ [N5: nat] : ( product_Pair @ nat @ nat @ N5 @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_6202_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,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X4 ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_6203_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_6204_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,Z2: $o] : Y5 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) )
@ ( ord_less_eq @ int ) ) ).
% less_eq_int.transfer
thf(fact_6205_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y4 @ V4 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_6206_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y4 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V4 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y4 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V4 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_6207_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_6208_intrel__iff,axiom,
! [X: nat,Y: nat,U: nat,V3: nat] :
( ( intrel @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ U @ V3 ) )
= ( ( plus_plus @ nat @ X @ V3 )
= ( plus_plus @ nat @ U @ Y ) ) ) ).
% intrel_iff
thf(fact_6209_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_6210_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,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X4 ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X4 ) ) ) ).
% uminus_int.rsp
thf(fact_6211_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_6212_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,Z2: $o] : Y5 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_6213_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y4 @ V4 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y4 @ V4 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_6214_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A6: A > B > $o,B6: 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 @ B6 @ 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 @ B6 ) @ A6 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_6215_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_6216_euclidean__size__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A] :
( ( ( euclid6346220572633701492n_size @ A @ B2 )
= ( zero_zero @ nat ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% euclidean_size_eq_0_iff
thf(fact_6217_size__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( zero_zero @ A ) )
= ( zero_zero @ nat ) ) ) ).
% size_0
thf(fact_6218_euclidean__size__greater__0__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
= ( B2
!= ( zero_zero @ A ) ) ) ) ).
% euclidean_size_greater_0_iff
thf(fact_6219_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,Z2: nat] : Y5 = Z2
@ ( size_size @ ( list @ A ) )
@ ( size_size @ ( list @ B ) ) ) ).
% length_transfer
thf(fact_6220_list__all2__lengthD,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs2: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P2 @ Xs2 @ Ys )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_6221_list__all2__append2,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Xs2: list @ A,Ys: list @ B,Zs: list @ B] :
( ( list_all2 @ A @ B @ P2 @ Xs2 @ ( append @ B @ Ys @ Zs ) )
= ( ? [Us3: list @ A,Vs3: list @ A] :
( ( Xs2
= ( append @ A @ Us3 @ Vs3 ) )
& ( ( size_size @ ( list @ A ) @ Us3 )
= ( size_size @ ( list @ B ) @ Ys ) )
& ( ( size_size @ ( list @ A ) @ Vs3 )
= ( size_size @ ( list @ B ) @ Zs ) )
& ( list_all2 @ A @ B @ P2 @ Us3 @ Ys )
& ( list_all2 @ A @ B @ P2 @ Vs3 @ Zs ) ) ) ) ).
% list_all2_append2
thf(fact_6222_list__all2__append1,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs2: list @ A,Ys: list @ A,Zs: list @ B] :
( ( list_all2 @ A @ B @ P2 @ ( append @ A @ Xs2 @ Ys ) @ Zs )
= ( ? [Us3: list @ B,Vs3: list @ B] :
( ( Zs
= ( append @ B @ Us3 @ Vs3 ) )
& ( ( size_size @ ( list @ B ) @ Us3 )
= ( size_size @ ( list @ A ) @ Xs2 ) )
& ( ( size_size @ ( list @ B ) @ Vs3 )
= ( size_size @ ( list @ A ) @ Ys ) )
& ( list_all2 @ A @ B @ P2 @ Xs2 @ Us3 )
& ( list_all2 @ A @ B @ P2 @ Ys @ Vs3 ) ) ) ) ).
% list_all2_append1
thf(fact_6223_list__all2__append,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys: list @ B,P2: A > B > $o,Us: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( list_all2 @ A @ B @ P2 @ ( append @ A @ Xs2 @ Us ) @ ( append @ B @ Ys @ Vs ) )
= ( ( list_all2 @ A @ B @ P2 @ Xs2 @ Ys )
& ( list_all2 @ A @ B @ P2 @ Us @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_6224_dvd__euclidean__size__eq__imp__dvd,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ) ).
% dvd_euclidean_size_eq_imp_dvd
thf(fact_6225_euclidean__size__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% euclidean_size_mult
thf(fact_6226_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs2: list @ A,Ys: list @ B,P: nat] :
( ( list_all2 @ A @ B @ P2 @ Xs2 @ Ys )
=> ( ( ord_less @ nat @ P @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P2 @ ( nth @ A @ Xs2 @ P ) @ ( nth @ B @ Ys @ P ) ) ) ) ).
% list_all2_nthD
thf(fact_6227_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs2: list @ A,Ys: list @ B,P: nat] :
( ( list_all2 @ A @ B @ P2 @ Xs2 @ Ys )
=> ( ( ord_less @ nat @ P @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( P2 @ ( nth @ A @ Xs2 @ P ) @ ( nth @ B @ Ys @ P ) ) ) ) ).
% list_all2_nthD2
thf(fact_6228_list__all2__all__nthI,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P2: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ A3 ) )
=> ( P2 @ ( nth @ A @ A3 @ N3 ) @ ( nth @ B @ B2 @ N3 ) ) )
=> ( list_all2 @ A @ B @ P2 @ A3 @ B2 ) ) ) ).
% list_all2_all_nthI
thf(fact_6229_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 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P4 @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys3 @ I ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_6230_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
= ( ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
& ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_iff_euclidean_size
thf(fact_6231_size__mult__mono_H,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B2 @ A3 ) ) ) ) ) ).
% size_mult_mono'
thf(fact_6232_size__mult__mono,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% size_mult_mono
thf(fact_6233_euclidean__size__times__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% euclidean_size_times_unit
thf(fact_6234_dvd__proper__imp__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ~ ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ) ).
% dvd_proper_imp_size_less
thf(fact_6235_dvd__imp__size__le,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ).
% dvd_imp_size_le
thf(fact_6236_mod__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% mod_size_less
thf(fact_6237_list__all2I,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P2: A > B > $o] :
( ! [X3: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ A3 @ B2 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P2 @ X3 ) )
=> ( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( list_all2 @ A @ B @ P2 @ A3 @ B2 ) ) ) ).
% list_all2I
thf(fact_6238_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_6239_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_6240_divmod__cases,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A3: A] :
( ( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( A3
!= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) )
=> ( ( ( B2
!= ( zero_zero @ A ) )
=> ! [Q3: A,R4: A] :
( ( ( euclid7384307370059645450egment @ A @ R4 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R4 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( R4
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= Q3 )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= R4 )
=> ( A3
!= ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R4 ) ) ) ) ) ) ) )
=> ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_6241_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_mult @ B )
& ( monoid_mult @ A ) )
=> ! [A6: A > B > $o] :
( ( A6 @ ( one_one @ A ) @ ( one_one @ 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 @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A6 ) @ A6 @ ( groups5270119922927024881d_list @ A ) @ ( groups5270119922927024881d_list @ B ) ) ) ) ) ).
% prod_list_transfer
thf(fact_6242_prod__list_OCons,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Xs2: list @ A] :
( ( groups5270119922927024881d_list @ A @ ( cons @ A @ X @ Xs2 ) )
= ( times_times @ A @ X @ ( groups5270119922927024881d_list @ A @ Xs2 ) ) ) ) ).
% prod_list.Cons
thf(fact_6243_prod__list_Oappend,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Xs2: list @ A,Ys: list @ A] :
( ( groups5270119922927024881d_list @ A @ ( append @ A @ Xs2 @ Ys ) )
= ( times_times @ A @ ( groups5270119922927024881d_list @ A @ Xs2 ) @ ( groups5270119922927024881d_list @ A @ Ys ) ) ) ) ).
% prod_list.append
thf(fact_6244_division__segment__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A3 ) ) )
= A3 ) ) ).
% division_segment_euclidean_size
thf(fact_6245_division__segment__not__0,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A] :
( ( euclid7384307370059645450egment @ A @ A3 )
!= ( zero_zero @ A ) ) ) ).
% division_segment_not_0
thf(fact_6246_division__segment__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ) ).
% division_segment_mult
thf(fact_6247_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_6248_division__segment__mod,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( euclid7384307370059645450egment @ A @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ).
% division_segment_mod
thf(fact_6249_prod__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A )
= ( ^ [Xs: list @ A] : ( foldr @ A @ A @ ( times_times @ A ) @ Xs @ ( one_one @ A ) ) ) ) ) ).
% prod_list.eq_foldr
thf(fact_6250_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( ( euclid7384307370059645450egment @ A @ A3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_6251_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R3: A,Q2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q2 @ B2 ) @ R3 ) @ B2 )
= Q2 ) ) ) ) ) ).
% div_bounded
thf(fact_6252_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R3: A,Q2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q2 @ B2 ) @ R3 )
= A3 )
=> ( ( divide_divide @ A @ A3 @ B2 )
= Q2 ) ) ) ) ) ) ).
% div_eqI
thf(fact_6253_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R3: A,Q2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q2 @ B2 ) @ R3 )
= A3 )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= R3 ) ) ) ) ) ) ).
% mod_eqI
thf(fact_6254_AboveS__def,axiom,
! [A: $tType] :
( ( order_AboveS @ A )
= ( ^ [R: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ( B5 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ B5 ) @ R ) ) ) ) ) ) ) ).
% AboveS_def
thf(fact_6255_less__eq__enat__def,axiom,
( ( ord_less_eq @ extended_enat )
= ( ^ [M6: extended_enat] :
( extended_case_enat @ $o
@ ^ [N1: nat] :
( extended_case_enat @ $o
@ ^ [M1: nat] : ( ord_less_eq @ nat @ M1 @ N1 )
@ $false
@ M6 )
@ $true ) ) ) ).
% less_eq_enat_def
thf(fact_6256_wo__rel_Osuc__greater,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B6: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R3 ) )
=> ( ( ( order_AboveS @ A @ R3 @ B6 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ B2 @ B6 )
=> ( ( ( bNF_Wellorder_wo_suc @ A @ R3 @ B6 )
!= B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B6 ) ) @ R3 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_6257_wo__rel_Oequals__suc__AboveS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B6: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ A3 @ ( order_AboveS @ A @ R3 @ B6 ) )
=> ( ! [A17: A] :
( ( member @ A @ A17 @ ( order_AboveS @ A @ R3 @ B6 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A17 ) @ R3 ) )
=> ( A3
= ( bNF_Wellorder_wo_suc @ A @ R3 @ B6 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_6258_wo__rel_Osuc__least__AboveS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B6: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( order_AboveS @ A @ R3 @ B6 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B6 ) @ A3 ) @ R3 ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_6259_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A6: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A6 )
=> ( ( ( order_AboveS @ A @ R3 @ A6 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R3 @ A6 ) ) @ R3 )
=> ( ( B2
!= ( bNF_Wellorder_wo_suc @ A @ R3 @ A6 ) )
=> ( member @ A @ B2 @ A6 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_6260_sorted__wrt__iff__nth__Suc__transp,axiom,
! [A: $tType,P2: A > A > $o,Xs2: list @ A] :
( ( transp @ A @ P2 )
=> ( ( sorted_wrt @ A @ P2 @ Xs2 )
= ( ! [I: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P2 @ ( nth @ A @ Xs2 @ I ) @ ( nth @ A @ Xs2 @ ( suc @ I ) ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
thf(fact_6261_transp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less_eq @ A ) ) ) ).
% transp_le
thf(fact_6262_transp__ge,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ).
% transp_ge
thf(fact_6263_bsqr__max2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A12: A,A23: A,B14: A,B23: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 @ A12 @ A23 ) @ ( product_Pair @ A @ A @ B14 @ B23 ) ) @ ( bNF_Wellorder_bsqr @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ B14 @ B23 ) ) @ R3 ) ) ) ).
% bsqr_max2
thf(fact_6264_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] :
( ! [A7: A,A14: A,B5: B] :
( ( Prod3 @ ( plus_plus @ A @ A7 @ A14 ) @ B5 )
= ( plus_plus @ C @ ( Prod3 @ A7 @ B5 ) @ ( Prod3 @ A14 @ B5 ) ) )
& ! [A7: A,B5: B,B12: B] :
( ( Prod3 @ A7 @ ( plus_plus @ B @ B5 @ B12 ) )
= ( plus_plus @ C @ ( Prod3 @ A7 @ B5 ) @ ( Prod3 @ A7 @ B12 ) ) )
& ! [R: real,A7: A,B5: B] :
( ( Prod3 @ ( real_V8093663219630862766scaleR @ A @ R @ A7 ) @ B5 )
= ( real_V8093663219630862766scaleR @ C @ R @ ( Prod3 @ A7 @ B5 ) ) )
& ! [A7: A,R: real,B5: B] :
( ( Prod3 @ A7 @ ( real_V8093663219630862766scaleR @ B @ R @ B5 ) )
= ( real_V8093663219630862766scaleR @ C @ R @ ( Prod3 @ A7 @ B5 ) ) )
& ? [K6: real] :
! [A7: A,B5: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod3 @ A7 @ B5 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A7 ) @ ( real_V7770717601297561774m_norm @ B @ B5 ) ) @ K6 ) ) ) ) ) ) ).
% bounded_bilinear_def
thf(fact_6265_bounded__bilinear_Ozero__left,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [Prod: A > B > C,B2: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( Prod @ ( zero_zero @ A ) @ B2 )
= ( zero_zero @ C ) ) ) ) ).
% bounded_bilinear.zero_left
thf(fact_6266_bounded__bilinear_Ozero__right,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [Prod: A > B > C,A3: A] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( Prod @ A3 @ ( zero_zero @ B ) )
= ( zero_zero @ C ) ) ) ) ).
% bounded_bilinear.zero_right
thf(fact_6267_bounded__bilinear__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ( real_V2442710119149674383linear @ A @ A @ A @ ( times_times @ A ) ) ) ).
% bounded_bilinear_mult
thf(fact_6268_well__order__on__domain,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_well_order_on @ A @ A6 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ A @ A3 @ A6 )
& ( member @ A @ B2 @ A6 ) ) ) ) ).
% well_order_on_domain
thf(fact_6269_natLeq__Well__order,axiom,
order_well_order_on @ nat @ ( field2 @ nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).
% natLeq_Well_order
thf(fact_6270_bounded__bilinear_Obounded,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ? [K9: real] :
! [A9: A,B7: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod @ A9 @ B7 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A9 ) @ ( real_V7770717601297561774m_norm @ B @ B7 ) ) @ K9 ) ) ) ) ).
% bounded_bilinear.bounded
thf(fact_6271_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,Y4: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y4 @ N )
& ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_6272_bounded__bilinear_Otendsto__right__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C,F3: D > B,F5: filter @ D,C3: A] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( filterlim @ D @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ D @ C
@ ^ [X4: D] : ( Prod @ C3 @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ) ).
% bounded_bilinear.tendsto_right_zero
thf(fact_6273_bounded__bilinear_Otendsto__left__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C,F3: D > A,F5: filter @ D,C3: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ C
@ ^ [X4: D] : ( Prod @ ( F3 @ X4 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ) ).
% bounded_bilinear.tendsto_left_zero
thf(fact_6274_bounded__bilinear_Otendsto__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C,F3: D > A,F5: filter @ D,G3: D > B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( ( filterlim @ D @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ D @ C
@ ^ [X4: D] : ( Prod @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ) ) ).
% bounded_bilinear.tendsto_zero
thf(fact_6275_bounded__bilinear_Ononneg__bounded,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ? [K9: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ K9 )
& ! [A9: A,B7: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod @ A9 @ B7 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A9 ) @ ( real_V7770717601297561774m_norm @ B @ B7 ) ) @ K9 ) ) ) ) ) ).
% bounded_bilinear.nonneg_bounded
thf(fact_6276_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,Y4: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y4 @ N )
& ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y4: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y4 @ N )
& ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_6277_Linear__order__Well__order__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R3 ) )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ! [Y4: A] :
( ( member @ A @ Y4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_6278_bounded__bilinear_Opos__bounded,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [A9: A,B7: B] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( Prod @ A9 @ B7 ) ) @ ( times_times @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A9 ) @ ( real_V7770717601297561774m_norm @ B @ B7 ) ) @ K9 ) ) ) ) ) ).
% bounded_bilinear.pos_bounded
thf(fact_6279_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,A17: A,B4: B] :
( ( Prod @ ( plus_plus @ A @ A5 @ A17 ) @ B4 )
= ( plus_plus @ C @ ( Prod @ A5 @ B4 ) @ ( Prod @ A17 @ B4 ) ) )
=> ( ! [A5: A,B4: B,B10: B] :
( ( Prod @ A5 @ ( plus_plus @ B @ B4 @ B10 ) )
= ( plus_plus @ C @ ( Prod @ A5 @ B4 ) @ ( Prod @ A5 @ B10 ) ) )
=> ( ! [R4: real,A5: A,B4: B] :
( ( Prod @ ( real_V8093663219630862766scaleR @ A @ R4 @ A5 ) @ B4 )
= ( real_V8093663219630862766scaleR @ C @ R4 @ ( Prod @ A5 @ B4 ) ) )
=> ( ! [A5: A,R4: real,B4: B] :
( ( Prod @ A5 @ ( real_V8093663219630862766scaleR @ B @ R4 @ B4 ) )
= ( real_V8093663219630862766scaleR @ C @ R4 @ ( 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_6280_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_6281_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_6282_bdd__above_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A,M5: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ M5 ) )
=> ( condit941137186595557371_above @ A @ A6 ) ) ) ).
% bdd_above.I
thf(fact_6283_bdd__above_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit941137186595557371_above @ A )
= ( ^ [A8: set @ A] :
? [M9: A] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ord_less_eq @ A @ X4 @ M9 ) ) ) ) ) ).
% bdd_above.unfold
thf(fact_6284_bdd__above_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A] :
( ( condit941137186595557371_above @ A @ A6 )
=> ~ ! [M8: A] :
~ ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ A @ X5 @ M8 ) ) ) ) ).
% bdd_above.E
thf(fact_6285_cSup__upper,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X8: set @ A] :
( ( member @ A @ X @ X8 )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% cSup_upper
thf(fact_6286_cSup__upper2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X8: set @ A,Y: A] :
( ( member @ A @ X @ X8 )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ord_less_eq @ A @ Y @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ).
% cSup_upper2
thf(fact_6287_bdd__above_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A6: set @ B,F3: B > A,M5: A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ M5 ) )
=> ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ).
% bdd_above.I2
thf(fact_6288_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A6: set @ B,X: B,U: A] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( member @ B @ X @ A6 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ X ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_6289_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: B,A6: set @ B,F3: B > A] :
( ( member @ B @ X @ A6 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ).
% cSUP_upper
thf(fact_6290_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B6: set @ A,A6: set @ A] :
( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A6 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ( ord_less_eq @ A @ B4 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B6 ) @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_6291_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_6292_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_6293_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,G3: C > A,B6: set @ C,F3: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ G3 @ B6 ) )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B6 )
& ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G3 @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_6294_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_6295_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,G3: B > A,B6: set @ B,F3: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G3 @ B6 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_6296_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ( condit941137186595557371_above @ A @ A6 )
=> ( ( condit941137186595557371_above @ A @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A6 @ B6 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_6297_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] :
! [Y4: A] :
( ( member @ A @ Y4 @ S3 )
=> ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_6298_Max_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ) ).
% Max.semilattice_order_set_axioms
thf(fact_6299_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_6300_bdd__belowI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A,M2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ M2 @ X3 ) )
=> ( condit1013018076250108175_below @ A @ A6 ) ) ) ).
% bdd_belowI
thf(fact_6301_bdd__below_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A,M5: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ M5 @ X3 ) )
=> ( condit1013018076250108175_below @ A @ A6 ) ) ) ).
% bdd_below.I
thf(fact_6302_bdd__below_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A6: set @ B,M5: A,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ M5 @ ( F3 @ X3 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ).
% bdd_below.I2
thf(fact_6303_bdd__belowI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A6: set @ B,M2: A,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ M2 @ ( F3 @ X3 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ).
% bdd_belowI2
thf(fact_6304_cInf__lower,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X8: set @ A] :
( ( member @ A @ X @ X8 )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X ) ) ) ) ).
% cInf_lower
thf(fact_6305_cInf__lower2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X8: set @ A,Y: A] :
( ( member @ A @ X @ X8 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y ) ) ) ) ) ).
% cInf_lower2
thf(fact_6306_bdd__below_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A] :
( ( condit1013018076250108175_below @ A @ A6 )
=> ~ ! [M8: A] :
~ ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( ord_less_eq @ A @ M8 @ X5 ) ) ) ) ).
% bdd_below.E
thf(fact_6307_bdd__below_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( ^ [A8: set @ A] :
? [M9: A] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ord_less_eq @ A @ M9 @ X4 ) ) ) ) ) ).
% bdd_below.unfold
thf(fact_6308_cINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A6: set @ B,X: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( member @ B @ X @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( F3 @ X ) ) ) ) ) ).
% cINF_lower
thf(fact_6309_cINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A6: set @ B,X: B,U: A] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( member @ B @ X @ A6 )
=> ( ( ord_less_eq @ A @ ( F3 @ X ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ U ) ) ) ) ) ).
% cINF_lower2
thf(fact_6310_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B6: set @ A,A6: set @ A] :
( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A6 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ( ord_less_eq @ A @ X5 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_6311_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_6312_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B6: set @ B,F3: C > A,A6: set @ C,G3: B > A] :
( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ F3 @ A6 ) )
=> ( ! [M: B] :
( ( member @ B @ M @ B6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ A6 )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ M ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6313_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_6314_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B6 ) @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_6315_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_6316_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_6317_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,G3: B > A,B6: set @ B,F3: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G3 @ B6 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B6 )
=> ( ord_less_eq @ A @ ( G3 @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6318_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ( condit1013018076250108175_below @ A @ A6 )
=> ( ( condit1013018076250108175_below @ A @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A6 @ B6 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B6 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_6319_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_6320_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_6321_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] :
! [Y4: A] :
( ( member @ A @ Y4 @ S3 )
=> ( ord_less_eq @ A @ X4 @ Y4 ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_6322_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_6323_Gcd__fin__def,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( bounde2362111253966948842tice_F @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ) ).
% Gcd_fin_def
thf(fact_6324_flip__pred,axiom,
! [A: $tType,B: $tType,A6: set @ ( product_prod @ A @ B ),R2: 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 @ R2 ) ) ) )
=> ( 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,Y4: B] : ( product_Pair @ B @ A @ Y4 @ X4 ) )
@ A6 )
@ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R2 ) ) ) ) ).
% flip_pred
thf(fact_6325_gen__length__def,axiom,
! [A: $tType] :
( ( gen_length @ A )
= ( ^ [N5: nat,Xs: list @ A] : ( plus_plus @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% gen_length_def
thf(fact_6326_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_6327_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_6328_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_6329_gen__length__code_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs2: list @ B] :
( ( gen_length @ B @ N @ ( cons @ B @ X @ Xs2 ) )
= ( gen_length @ B @ ( suc @ N ) @ Xs2 ) ) ).
% gen_length_code(2)
thf(fact_6330_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_6331_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_6332_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] :
? [A13: B,A24: B] :
( ( Uu3
= ( product_Pair @ A @ A @ ( F4 @ A13 ) @ ( F4 @ A24 ) ) )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A13 @ A24 ) @ R6 ) ) ) ) ) ).
% relImage_def
thf(fact_6333_card__def,axiom,
! [B: $tType] :
( ( finite_card @ B )
= ( finite_folding_F @ B @ nat
@ ^ [Uu3: B] : suc
@ ( zero_zero @ nat ) ) ) ).
% card_def
thf(fact_6334_mono__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ D )
& ( order @ C )
& ( order @ A ) )
=> ! [A6: A > B > $o,B6: 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,Z2: $o] : Y5 = Z2 )
@ ( ord_less_eq @ A )
@ ( ord_less_eq @ B ) )
=> ( ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B6
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B6
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( ord_less_eq @ C )
@ ( ord_less_eq @ D ) )
=> ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A6 @ B6 )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ ( order_mono @ A @ C )
@ ( order_mono @ B @ D ) ) ) ) ) ) ).
% mono_transfer
thf(fact_6335_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,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) ) ) ) ).
% plus_rat_def
thf(fact_6336_diff__rat,axiom,
! [B2: int,D3: int,A3: int,C3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( minus_minus @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C3 @ D3 ) )
= ( fract @ ( minus_minus @ int @ ( times_times @ int @ A3 @ D3 ) @ ( times_times @ int @ C3 @ B2 ) ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ).
% diff_rat
thf(fact_6337_sgn__rat,axiom,
! [A3: int,B2: int] :
( ( sgn_sgn @ rat @ ( fract @ A3 @ B2 ) )
= ( ring_1_of_int @ rat @ ( times_times @ int @ ( sgn_sgn @ int @ A3 ) @ ( sgn_sgn @ int @ B2 ) ) ) ) ).
% sgn_rat
thf(fact_6338_mult__rat,axiom,
! [A3: int,B2: int,C3: int,D3: int] :
( ( times_times @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C3 @ D3 ) )
= ( fract @ ( times_times @ int @ A3 @ C3 ) @ ( times_times @ int @ B2 @ D3 ) ) ) ).
% mult_rat
thf(fact_6339_divide__rat,axiom,
! [A3: int,B2: int,C3: int,D3: int] :
( ( divide_divide @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C3 @ D3 ) )
= ( fract @ ( times_times @ int @ A3 @ D3 ) @ ( times_times @ int @ B2 @ C3 ) ) ) ).
% divide_rat
thf(fact_6340_less__rat,axiom,
! [B2: int,D3: int,A3: int,C3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C3 @ D3 ) )
= ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D3 ) @ ( times_times @ int @ B2 @ D3 ) ) @ ( times_times @ int @ ( times_times @ int @ C3 @ B2 ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ) ).
% less_rat
thf(fact_6341_add__rat,axiom,
! [B2: int,D3: int,A3: int,C3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( plus_plus @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C3 @ D3 ) )
= ( fract @ ( plus_plus @ int @ ( times_times @ int @ A3 @ D3 ) @ ( times_times @ int @ C3 @ B2 ) ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ).
% add_rat
thf(fact_6342_le__rat,axiom,
! [B2: int,D3: int,A3: int,C3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C3 @ D3 ) )
= ( ord_less_eq @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D3 ) @ ( times_times @ int @ B2 @ D3 ) ) @ ( times_times @ int @ ( times_times @ int @ C3 @ B2 ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ) ).
% le_rat
thf(fact_6343_mult__rat__cancel,axiom,
! [C3: int,A3: int,B2: int] :
( ( C3
!= ( zero_zero @ int ) )
=> ( ( fract @ ( times_times @ int @ C3 @ A3 ) @ ( times_times @ int @ C3 @ B2 ) )
= ( fract @ A3 @ B2 ) ) ) ).
% mult_rat_cancel
thf(fact_6344_eq__rat_I1_J,axiom,
! [B2: int,D3: int,A3: int,C3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( ( fract @ A3 @ B2 )
= ( fract @ C3 @ D3 ) )
= ( ( times_times @ int @ A3 @ D3 )
= ( times_times @ int @ C3 @ B2 ) ) ) ) ) ).
% eq_rat(1)
thf(fact_6345_positive__rat,axiom,
! [A3: int,B2: int] :
( ( positive @ ( fract @ A3 @ B2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ A3 @ B2 ) ) ) ).
% positive_rat
thf(fact_6346_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,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) ) ) ) ).
% times_rat_def
thf(fact_6347_irrefl__tranclI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A] :
( ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R3 ) @ ( transitive_rtrancl @ A @ R3 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R3 ) ) ) ).
% irrefl_tranclI
thf(fact_6348_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= ( insert @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) ) ).
% prod_set_simps(2)
thf(fact_6349_converse__iff,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R3 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R3 ) ) ).
% converse_iff
thf(fact_6350_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( conversep @ A @ B
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R3 ) )
= ( ^ [X4: B,Y4: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y4 ) @ ( converse @ A @ B @ R3 ) ) ) ) ).
% conversep_converse_eq
thf(fact_6351_converse__unfold,axiom,
! [A: $tType,B: $tType] :
( ( converse @ B @ A )
= ( ^ [R: set @ ( product_prod @ B @ A )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [Y4: A,X4: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y4 ) @ R ) ) ) ) ) ).
% converse_unfold
thf(fact_6352_trancl__converseI,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R3 ) ) ) ) ).
% trancl_converseI
thf(fact_6353_trancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_converseD
thf(fact_6354_converseI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ ( converse @ A @ B @ R3 ) ) ) ).
% converseI
thf(fact_6355_converseE,axiom,
! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R3 ) )
=> ~ ! [X3: B,Y3: A] :
( ( Yx
= ( product_Pair @ A @ B @ Y3 @ X3 ) )
=> ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ R3 ) ) ) ).
% converseE
thf(fact_6356_converseD,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R3 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R3 ) ) ).
% converseD
thf(fact_6357_converse_Osimps,axiom,
! [B: $tType,A: $tType,A12: B,A23: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A12 @ A23 ) @ ( converse @ A @ B @ R3 ) )
= ( ? [A7: A,B5: B] :
( ( A12 = B5 )
& ( A23 = A7 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R3 ) ) ) ) ).
% converse.simps
thf(fact_6358_converse_Ocases,axiom,
! [B: $tType,A: $tType,A12: B,A23: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A12 @ A23 ) @ ( converse @ A @ B @ R3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A23 @ A12 ) @ R3 ) ) ).
% converse.cases
thf(fact_6359_rtrancl__converseI,axiom,
! [A: $tType,Y: A,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R3 ) ) ) ) ).
% rtrancl_converseI
thf(fact_6360_rtrancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ).
% rtrancl_converseD
thf(fact_6361_converse__def,axiom,
! [B: $tType,A: $tType] :
( ( converse @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ( conversep @ A @ B
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R ) ) ) ) ) ) ).
% converse_def
thf(fact_6362_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X: A,Y: B] :
( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% prod_set_simps(1)
thf(fact_6363_integer__of__char__code,axiom,
! [B0: $o,B14: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( integer_of_char @ ( char2 @ B0 @ B14 @ B23 @ 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 @ B23 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B14 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B0 ) ) ) ).
% integer_of_char_code
thf(fact_6364_char_Osize_I2_J,axiom,
! [X1: $o,X2: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size @ char @ ( char2 @ X1 @ X2 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size(2)
thf(fact_6365_char_Osize__gen,axiom,
! [X1: $o,X2: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_char @ ( char2 @ X1 @ X2 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size_gen
thf(fact_6366_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_6367_idiff__enat__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( extended_enat2 @ ( zero_zero @ nat ) ) )
= N ) ).
% idiff_enat_0_right
thf(fact_6368_idiff__enat__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ ( zero_zero @ nat ) ) @ N )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% idiff_enat_0
thf(fact_6369_times__enat__simps_I1_J,axiom,
! [M2: nat,N: nat] :
( ( times_times @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( times_times @ nat @ M2 @ N ) ) ) ).
% times_enat_simps(1)
thf(fact_6370_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_6371_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_6372_enat__0__iff_I2_J,axiom,
! [X: nat] :
( ( ( zero_zero @ extended_enat )
= ( extended_enat2 @ X ) )
= ( X
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(2)
thf(fact_6373_enat__0__iff_I1_J,axiom,
! [X: nat] :
( ( ( extended_enat2 @ X )
= ( zero_zero @ extended_enat ) )
= ( X
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(1)
thf(fact_6374_zero__enat__def,axiom,
( ( zero_zero @ extended_enat )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% zero_enat_def
thf(fact_6375_iadd__le__enat__iff,axiom,
! [X: extended_enat,Y: extended_enat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( plus_plus @ extended_enat @ X @ Y ) @ ( extended_enat2 @ N ) )
= ( ? [Y8: nat,X9: nat] :
( ( X
= ( extended_enat2 @ X9 ) )
& ( Y
= ( extended_enat2 @ Y8 ) )
& ( ord_less_eq @ nat @ ( plus_plus @ nat @ X9 @ Y8 ) @ N ) ) ) ) ).
% iadd_le_enat_iff
thf(fact_6376_elimnum,axiom,
! [Info2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) ) ).
% elimnum
thf(fact_6377_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
! [Info2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,L: nat] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ L ) )
= ( vEBT_Node @ Info2 @ 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 ) ) ) ) ) )
@ TreeList2 ) )
@ ( 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_6378_enat__0__less__mult__iff,axiom,
! [M2: extended_enat,N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( times_times @ extended_enat @ M2 @ N ) )
= ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ M2 )
& ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ) ) ).
% enat_0_less_mult_iff
thf(fact_6379_imult__is__0,axiom,
! [M2: extended_enat,N: extended_enat] :
( ( ( times_times @ extended_enat @ M2 @ N )
= ( zero_zero @ extended_enat ) )
= ( ( M2
= ( zero_zero @ extended_enat ) )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% imult_is_0
thf(fact_6380_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
! [A3: $o,B2: $o,Uu: extended_enat] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Leaf @ A3 @ B2 ) @ Uu )
= ( vEBT_Leaf @ A3 @ B2 ) ) ).
% VEBT_internal.elim_dead.simps(1)
thf(fact_6381_VEBT__internal_Oelim__dead_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_elim_dead @ X @ Xa2 )
= Y )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A5 @ B4 ) )
=> ( Y
!= ( vEBT_Leaf @ A5 @ B4 ) ) )
=> ( ! [Info: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( Xa2
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( Y
!= ( vEBT_Node @ Info @ 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 ) ) ) ) ) )
@ TreeList )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) )
=> ~ ! [Info: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
=> ! [L4: nat] :
( ( Xa2
= ( extended_enat2 @ L4 ) )
=> ( Y
!= ( vEBT_Node @ Info @ 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 ) ) ) ) ) )
@ TreeList ) )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( 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_6382_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
! [Info2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( vEBT_Node @ Info2 @ 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 ) ) ) ) ) )
@ TreeList2 )
@ ( vEBT_VEBT_elim_dead @ Summary @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% VEBT_internal.elim_dead.simps(2)
thf(fact_6383_elimcomplete,axiom,
! [Info2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) ) ).
% elimcomplete
thf(fact_6384_times__enat__simps_I2_J,axiom,
( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% times_enat_simps(2)
thf(fact_6385_times__enat__simps_I3_J,axiom,
! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ N ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ N ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% times_enat_simps(3)
thf(fact_6386_times__enat__simps_I4_J,axiom,
! [M2: nat] :
( ( ( M2
= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( M2
!= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% times_enat_simps(4)
thf(fact_6387_imult__is__infinity,axiom,
! [A3: extended_enat,B2: extended_enat] :
( ( ( times_times @ extended_enat @ A3 @ B2 )
= ( extend4730790105801354508finity @ extended_enat ) )
= ( ( ( A3
= ( extend4730790105801354508finity @ extended_enat ) )
& ( B2
!= ( zero_zero @ extended_enat ) ) )
| ( ( B2
= ( extend4730790105801354508finity @ extended_enat ) )
& ( A3
!= ( zero_zero @ extended_enat ) ) ) ) ) ).
% imult_is_infinity
thf(fact_6388_VEBT__internal_Oelim__dead_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ extended_enat] :
( ! [A5: $o,B4: $o,Uu2: extended_enat] :
( X
!= ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Leaf @ A5 @ B4 ) @ Uu2 ) )
=> ( ! [Info: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
!= ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ ( extend4730790105801354508finity @ extended_enat ) ) )
=> ~ ! [Info: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT,L4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ).
% VEBT_internal.elim_dead.cases
thf(fact_6389_imult__infinity,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ N )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% imult_infinity
thf(fact_6390_imult__infinity__right,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
=> ( ( times_times @ extended_enat @ N @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% imult_infinity_right
thf(fact_6391_times__enat__def,axiom,
( ( times_times @ extended_enat )
= ( ^ [M6: extended_enat,N5: extended_enat] :
( extended_case_enat @ extended_enat
@ ^ [O: nat] :
( extended_case_enat @ extended_enat
@ ^ [P5: nat] : ( extended_enat2 @ ( times_times @ nat @ O @ P5 ) )
@ ( if @ extended_enat
@ ( O
= ( zero_zero @ nat ) )
@ ( zero_zero @ extended_enat )
@ ( extend4730790105801354508finity @ extended_enat ) )
@ N5 )
@ ( if @ extended_enat
@ ( N5
= ( zero_zero @ extended_enat ) )
@ ( zero_zero @ extended_enat )
@ ( extend4730790105801354508finity @ extended_enat ) )
@ M6 ) ) ) ).
% times_enat_def
thf(fact_6392_VEBT__internal_Oelim__dead_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_elim_dead @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ X @ Xa2 ) )
=> ( ! [A5: $o,B4: $o] :
( ( X
= ( 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 ) ) ) )
=> ( ! [Info: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
=> ( ( Xa2
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ( Y
= ( vEBT_Node @ Info @ 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 ) ) ) ) ) )
@ TreeList )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extend4730790105801354508finity @ extended_enat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) )
=> ~ ! [Info: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
=> ! [L4: nat] :
( ( Xa2
= ( extended_enat2 @ L4 ) )
=> ( ( Y
= ( vEBT_Node @ Info @ 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 ) ) ) ) ) )
@ TreeList ) )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( 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 @ Info @ Deg2 @ TreeList @ Summary2 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.pelims
thf(fact_6393_eSuc__def,axiom,
( extended_eSuc
= ( extended_case_enat @ extended_enat
@ ^ [N5: nat] : ( extended_enat2 @ ( suc @ N5 ) )
@ ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% eSuc_def
thf(fact_6394_binomial__def,axiom,
( binomial
= ( ^ [N5: nat,K3: nat] :
( finite_card @ ( set @ nat )
@ ( collect @ ( set @ nat )
@ ^ [K6: set @ nat] :
( ( member @ ( set @ nat ) @ K6 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) )
& ( ( finite_card @ nat @ K6 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_6395_enat__eSuc__iff,axiom,
! [Y: nat,X: extended_enat] :
( ( ( extended_enat2 @ Y )
= ( extended_eSuc @ X ) )
= ( ? [N5: nat] :
( ( Y
= ( suc @ N5 ) )
& ( ( extended_enat2 @ N5 )
= X ) ) ) ) ).
% enat_eSuc_iff
thf(fact_6396_eSuc__enat__iff,axiom,
! [X: extended_enat,Y: nat] :
( ( ( extended_eSuc @ X )
= ( extended_enat2 @ Y ) )
= ( ? [N5: nat] :
( ( Y
= ( suc @ N5 ) )
& ( X
= ( extended_enat2 @ N5 ) ) ) ) ) ).
% eSuc_enat_iff
thf(fact_6397_eSuc__enat,axiom,
! [N: nat] :
( ( extended_eSuc @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( suc @ N ) ) ) ).
% eSuc_enat
thf(fact_6398_mult__eSuc__right,axiom,
! [M2: extended_enat,N: extended_enat] :
( ( times_times @ extended_enat @ M2 @ ( extended_eSuc @ N ) )
= ( plus_plus @ extended_enat @ M2 @ ( times_times @ extended_enat @ M2 @ N ) ) ) ).
% mult_eSuc_right
thf(fact_6399_mult__eSuc,axiom,
! [M2: extended_enat,N: extended_enat] :
( ( times_times @ extended_enat @ ( extended_eSuc @ M2 ) @ N )
= ( plus_plus @ extended_enat @ N @ ( times_times @ extended_enat @ M2 @ N ) ) ) ).
% mult_eSuc
thf(fact_6400_Nats__altdef2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [N5: A] :
( ( member @ A @ N5 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ N5 ) ) ) ) ) ).
% Nats_altdef2
thf(fact_6401_less__than__iff,axiom,
! [X: nat,Y: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ less_than )
= ( ord_less @ nat @ X @ Y ) ) ).
% less_than_iff
thf(fact_6402_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_6403_Nats__induct,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A,P2: A > $o] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ( ! [N3: nat] : ( P2 @ ( semiring_1_of_nat @ A @ N3 ) )
=> ( P2 @ X ) ) ) ) ).
% Nats_induct
thf(fact_6404_Nats__cases,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ~ ! [N3: nat] :
( X
!= ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% Nats_cases
thf(fact_6405_Nats__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_1
thf(fact_6406_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_6407_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_6408_Nats__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_0
thf(fact_6409_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_6410_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_6411_lenlex__def,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ ( list @ A ) ) @ ( list @ A ) @ ( lex_prod @ nat @ ( list @ A ) @ less_than @ ( lex @ A @ R ) )
@ ^ [Xs: list @ A] : ( product_Pair @ nat @ ( list @ A ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs ) ) ) ) ).
% lenlex_def
thf(fact_6412_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_6413_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R3: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R3 @ F3 ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) @ R3 ) ) ).
% in_inv_image
thf(fact_6414_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) @ R ) ) ) ) ) ).
% inv_image_def
thf(fact_6415_Real_Opositive_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ $o @ $o @ realrel
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ ^ [X7: nat > rat] :
? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( X7 @ N5 ) ) ) )
@ ^ [X7: nat > rat] :
? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( X7 @ N5 ) ) ) ) ) ).
% Real.positive.rsp
thf(fact_6416_Gcd__nat__set__eq__fold,axiom,
! [Xs2: list @ nat] :
( ( gcd_Gcd @ nat @ ( set2 @ nat @ Xs2 ) )
= ( fold @ nat @ nat @ ( gcd_gcd @ nat ) @ Xs2 @ ( zero_zero @ nat ) ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_6417_times__real_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ realrel @ ( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ realrel @ realrel )
@ ^ [X7: nat > rat,Y10: nat > rat,N5: nat] : ( times_times @ rat @ ( X7 @ N5 ) @ ( Y10 @ N5 ) )
@ ^ [X7: nat > rat,Y10: nat > rat,N5: nat] : ( times_times @ rat @ ( X7 @ N5 ) @ ( Y10 @ N5 ) ) ) ).
% times_real.rsp
thf(fact_6418_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_6419_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_6420_of__rat__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ rat @ R3 @ ( zero_zero @ rat ) ) ) ) ).
% of_rat_le_0_iff
thf(fact_6421_zero__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R3 ) )
= ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ R3 ) ) ) ).
% zero_le_of_rat_iff
thf(fact_6422_zero__eq__of__rat__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( zero_zero @ A )
= ( field_char_0_of_rat @ A @ A3 ) )
= ( ( zero_zero @ rat )
= A3 ) ) ) ).
% zero_eq_of_rat_iff
thf(fact_6423_of__rat__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( field_char_0_of_rat @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ rat ) ) ) ) ).
% of_rat_eq_0_iff
thf(fact_6424_of__rat__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A @ ( zero_zero @ rat ) )
= ( zero_zero @ A ) ) ) ).
% of_rat_0
thf(fact_6425_of__rat__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ rat @ R3 @ ( one_one @ rat ) ) ) ) ).
% of_rat_le_1_iff
thf(fact_6426_one__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R3 ) )
= ( ord_less_eq @ rat @ ( one_one @ rat ) @ R3 ) ) ) ).
% one_le_of_rat_iff
thf(fact_6427_zero__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R3 ) )
= ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 ) ) ) ).
% zero_less_of_rat_iff
thf(fact_6428_of__rat__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( zero_zero @ A ) )
= ( ord_less @ rat @ R3 @ ( zero_zero @ rat ) ) ) ) ).
% of_rat_less_0_iff
thf(fact_6429_of__rat__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat,B2: rat] :
( ( field_char_0_of_rat @ A @ ( times_times @ rat @ A3 @ B2 ) )
= ( times_times @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% of_rat_mult
thf(fact_6430_of__rat__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat,S2: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( field_char_0_of_rat @ A @ S2 ) )
= ( ord_less_eq @ rat @ R3 @ S2 ) ) ) ).
% of_rat_less_eq
thf(fact_6431_num_Orec__transfer,axiom,
! [A: $tType,B: $tType,S3: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( num > A > A ) > ( num > A > A ) > num > A ) @ ( ( num > B > B ) > ( num > B > B ) > num > B ) @ S3
@ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( ( num > A > A ) > num > A ) @ ( ( num > B > B ) > num > B )
@ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
@ ^ [Y5: num,Z2: num] : Y5 = Z2
@ ( bNF_rel_fun @ A @ B @ A @ B @ S3 @ S3 ) )
@ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( num > A ) @ ( num > B )
@ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
@ ^ [Y5: num,Z2: num] : Y5 = Z2
@ ( bNF_rel_fun @ A @ B @ A @ B @ S3 @ S3 ) )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y5: num,Z2: num] : Y5 = Z2
@ S3 ) ) )
@ ( rec_num @ A )
@ ( rec_num @ B ) ) ).
% num.rec_transfer
thf(fact_6432_Real_Opositive_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ $o @ $o @ pcr_real
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ ^ [X7: nat > rat] :
? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( X7 @ N5 ) ) ) )
@ positive2 ) ).
% Real.positive.transfer
thf(fact_6433_Real_Opositive__mult,axiom,
! [X: real,Y: real] :
( ( positive2 @ X )
=> ( ( positive2 @ Y )
=> ( positive2 @ ( times_times @ real @ X @ Y ) ) ) ) ).
% Real.positive_mult
thf(fact_6434_times__real_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ ( ( nat > rat ) > nat > rat ) @ ( real > real ) @ pcr_real @ ( bNF_rel_fun @ ( nat > rat ) @ real @ ( nat > rat ) @ real @ pcr_real @ pcr_real )
@ ^ [X7: nat > rat,Y10: nat > rat,N5: nat] : ( times_times @ rat @ ( X7 @ N5 ) @ ( Y10 @ N5 ) )
@ ( times_times @ real ) ) ).
% times_real.transfer
thf(fact_6435_num_Ocase__transfer,axiom,
! [A: $tType,B: $tType,S3: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( num > A ) > ( num > A ) > num > A ) @ ( ( num > B ) > ( num > B ) > num > B ) @ S3
@ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( ( num > A ) > num > A ) @ ( ( num > B ) > num > B )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y5: num,Z2: num] : Y5 = Z2
@ S3 )
@ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( num > A ) @ ( num > B )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y5: num,Z2: num] : Y5 = Z2
@ S3 )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y5: num,Z2: num] : Y5 = Z2
@ S3 ) ) )
@ ( case_num @ A )
@ ( case_num @ B ) ) ).
% num.case_transfer
thf(fact_6436_Real_Opositive_Orep__eq,axiom,
( positive2
= ( ^ [X4: real] :
? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( rep_real @ X4 @ N5 ) ) ) ) ) ) ).
% Real.positive.rep_eq
thf(fact_6437_num_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F22: num > A,F32: num > A,Num: num] :
( ( H @ ( case_num @ A @ F1 @ F22 @ F32 @ Num ) )
= ( case_num @ B @ ( H @ F1 )
@ ^ [X4: num] : ( H @ ( F22 @ X4 ) )
@ ^ [X4: num] : ( H @ ( F32 @ X4 ) )
@ Num ) ) ).
% num.case_distrib
thf(fact_6438_Real_Opositive__def,axiom,
( positive2
= ( map_fun @ real @ ( nat > rat ) @ $o @ $o @ rep_real @ ( id @ $o )
@ ^ [X7: nat > rat] :
? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( X7 @ N5 ) ) ) ) ) ) ).
% Real.positive_def
thf(fact_6439_Real_Opositive_Oabs__eq,axiom,
! [X: nat > rat] :
( ( realrel @ X @ X )
=> ( ( positive2 @ ( real2 @ X ) )
= ( ? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( X @ N5 ) ) ) ) ) ) ) ).
% Real.positive.abs_eq
thf(fact_6440_times__real_Oabs__eq,axiom,
! [Xa2: nat > rat,X: nat > rat] :
( ( realrel @ Xa2 @ Xa2 )
=> ( ( realrel @ X @ X )
=> ( ( times_times @ real @ ( real2 @ Xa2 ) @ ( real2 @ X ) )
= ( real2
@ ^ [N5: nat] : ( times_times @ rat @ ( Xa2 @ N5 ) @ ( X @ N5 ) ) ) ) ) ) ).
% times_real.abs_eq
thf(fact_6441_le__Real,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( ord_less_eq @ real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
= ( ! [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
=> ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less_eq @ rat @ ( X8 @ N5 ) @ ( plus_plus @ rat @ ( Y7 @ N5 ) @ R ) ) ) ) ) ) ) ) ).
% le_Real
thf(fact_6442_not__positive__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( ~ ( positive2 @ ( real2 @ X8 ) ) )
= ( ! [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
=> ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less_eq @ rat @ ( X8 @ N5 ) @ R ) ) ) ) ) ) ).
% not_positive_Real
thf(fact_6443_cauchy__mult,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( cauchy
@ ^ [N5: nat] : ( times_times @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ).
% cauchy_mult
thf(fact_6444_mult__Real,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( times_times @ real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
= ( real2
@ ^ [N5: nat] : ( times_times @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ) ).
% mult_Real
thf(fact_6445_cauchyD,axiom,
! [X8: nat > rat,R3: rat] :
( ( cauchy @ X8 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 )
=> ? [K: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ K @ M3 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ K @ N4 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X8 @ M3 ) @ ( X8 @ N4 ) ) ) @ R3 ) ) ) ) ) ).
% cauchyD
thf(fact_6446_cauchyI,axiom,
! [X8: nat > rat] :
( ! [R4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R4 )
=> ? [K4: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ K4 @ M )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ K4 @ N3 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X8 @ M ) @ ( X8 @ N3 ) ) ) @ R4 ) ) ) )
=> ( cauchy @ X8 ) ) ).
% cauchyI
thf(fact_6447_cauchy__def,axiom,
( cauchy
= ( ^ [X7: nat > rat] :
! [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
=> ? [K3: nat] :
! [M6: nat] :
( ( ord_less_eq @ nat @ K3 @ M6 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X7 @ M6 ) @ ( X7 @ N5 ) ) ) @ R ) ) ) ) ) ) ).
% cauchy_def
thf(fact_6448_positive__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( positive2 @ ( real2 @ X8 ) )
= ( ? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( X8 @ N5 ) ) ) ) ) ) ) ).
% positive_Real
thf(fact_6449_cauchy__not__vanishes__cases,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ? [B4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B4 )
& ? [K: nat] :
( ! [N4: nat] :
( ( ord_less_eq @ nat @ K @ N4 )
=> ( ord_less @ rat @ B4 @ ( uminus_uminus @ rat @ ( X8 @ N4 ) ) ) )
| ! [N4: nat] :
( ( ord_less_eq @ nat @ K @ N4 )
=> ( ord_less @ rat @ B4 @ ( X8 @ N4 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes_cases
thf(fact_6450_cauchy__not__vanishes,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ? [B4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B4 )
& ? [K: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ K @ N4 )
=> ( ord_less @ rat @ B4 @ ( abs_abs @ rat @ ( X8 @ N4 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes
thf(fact_6451_vanishesD,axiom,
! [X8: nat > rat,R3: rat] :
( ( vanishes @ X8 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 )
=> ? [K: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ K @ N4 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X8 @ N4 ) ) @ R3 ) ) ) ) ).
% vanishesD
thf(fact_6452_vanishesI,axiom,
! [X8: nat > rat] :
( ! [R4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R4 )
=> ? [K4: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K4 @ N3 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X8 @ N3 ) ) @ R4 ) ) )
=> ( vanishes @ X8 ) ) ).
% vanishesI
thf(fact_6453_vanishes__def,axiom,
( vanishes
= ( ^ [X7: nat > rat] :
! [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
=> ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X7 @ N5 ) ) @ R ) ) ) ) ) ).
% vanishes_def
thf(fact_6454_vanishes__mult__bounded,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ? [A9: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ A9 )
& ! [N3: nat] : ( ord_less @ rat @ ( abs_abs @ rat @ ( X8 @ N3 ) ) @ A9 ) )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N5: nat] : ( times_times @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ).
% vanishes_mult_bounded
thf(fact_6455_surj__int__decode,axiom,
( ( image2 @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ int ) ) ) ).
% surj_int_decode
thf(fact_6456_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,V4: C] : ( product_Pair @ A @ B @ K3 @ ( F3 @ V4 ) ) )
@ Xs2 ) )
= ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F3 ) @ ( map_of @ A @ C @ Xs2 ) ) ) ).
% map_of_map
thf(fact_6457_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_6458_None__eq__map__option__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,X: option @ B] :
( ( ( none @ A )
= ( map_option @ B @ A @ F3 @ X ) )
= ( X
= ( none @ B ) ) ) ).
% None_eq_map_option_iff
thf(fact_6459_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_6460_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_6461_case__map__option,axiom,
! [B: $tType,A: $tType,C: $tType,G3: A,H: B > A,F3: C > B,X: option @ C] :
( ( case_option @ A @ B @ G3 @ H @ ( map_option @ C @ B @ F3 @ X ) )
= ( case_option @ A @ C @ G3 @ ( comp @ B @ A @ C @ H @ F3 ) @ X ) ) ).
% case_map_option
thf(fact_6462_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_6463_bij__int__decode,axiom,
bij_betw @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ int ) ) ).
% bij_int_decode
thf(fact_6464_inj__int__decode,axiom,
! [A6: set @ nat] : ( inj_on @ nat @ int @ nat_int_decode @ A6 ) ).
% inj_int_decode
thf(fact_6465_option_Omap__id,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% option.map_id
thf(fact_6466_option_Omap__id0,axiom,
! [A: $tType] :
( ( map_option @ A @ A @ ( id @ A ) )
= ( id @ ( option @ A ) ) ) ).
% option.map_id0
thf(fact_6467_map__option_Oidentity,axiom,
! [A: $tType] :
( ( map_option @ A @ A
@ ^ [X4: A] : X4 )
= ( id @ ( option @ A ) ) ) ).
% map_option.identity
thf(fact_6468_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_6469_option_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G3: B > C,F3: A > B,V3: option @ A] :
( ( map_option @ B @ C @ G3 @ ( map_option @ A @ B @ F3 @ V3 ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ G3 @ F3 ) @ V3 ) ) ).
% option.map_comp
thf(fact_6470_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_6471_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_6472_option_Omap__ident,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A
@ ^ [X4: A] : X4
@ T2 )
= T2 ) ).
% option.map_ident
thf(fact_6473_int__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( ( nat_int_decode @ X )
= ( nat_int_decode @ Y ) )
= ( X = Y ) ) ).
% int_decode_eq
thf(fact_6474_map__option__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F3: A > B,G3: A > B] :
( ( X = Y )
=> ( ! [A5: A] :
( ( Y
= ( some @ A @ A5 ) )
=> ( ( F3 @ A5 )
= ( G3 @ A5 ) ) )
=> ( ( map_option @ A @ B @ F3 @ X )
= ( map_option @ A @ B @ G3 @ Y ) ) ) ) ).
% map_option_cong
thf(fact_6475_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_6476_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_6477_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_6478_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_6479_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_6480_nat__to__rat__surj__def,axiom,
( nat_to_rat_surj
= ( ^ [N5: nat] :
( product_case_prod @ nat @ nat @ rat
@ ^ [A7: nat,B5: nat] : ( fract @ ( nat_int_decode @ A7 ) @ ( nat_int_decode @ B5 ) )
@ ( nat_prod_decode @ N5 ) ) ) ) ).
% nat_to_rat_surj_def
thf(fact_6481_aboveS__def,axiom,
! [A: $tType] :
( ( order_aboveS @ A )
= ( ^ [R: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B5: A] :
( ( B5 != A7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R ) ) ) ) ) ).
% aboveS_def
thf(fact_6482_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_6483_bij__int__encode,axiom,
bij_betw @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_int_encode
thf(fact_6484_int__encode__inverse,axiom,
! [X: int] :
( ( nat_int_decode @ ( nat_int_encode @ X ) )
= X ) ).
% int_encode_inverse
thf(fact_6485_int__decode__inverse,axiom,
! [N: nat] :
( ( nat_int_encode @ ( nat_int_decode @ N ) )
= N ) ).
% int_decode_inverse
thf(fact_6486_inj__int__encode,axiom,
! [A6: set @ int] : ( inj_on @ int @ nat @ nat_int_encode @ A6 ) ).
% inj_int_encode
thf(fact_6487_int__encode__eq,axiom,
! [X: int,Y: int] :
( ( ( nat_int_encode @ X )
= ( nat_int_encode @ Y ) )
= ( X = Y ) ) ).
% int_encode_eq
thf(fact_6488_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_6489_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_6490_surj__int__encode,axiom,
( ( image2 @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_int_encode
thf(fact_6491_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr7122648621184425601vImage @ A @ B )
= ( ^ [A8: set @ A,R6: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [A13: A,A24: A] :
( ( Uu3
= ( product_Pair @ A @ A @ A13 @ A24 ) )
& ( member @ A @ A13 @ A8 )
& ( member @ A @ A24 @ A8 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ A13 ) @ ( F4 @ A24 ) ) @ R6 ) ) ) ) ) ).
% relInvImage_def
thf(fact_6492_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_6493_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_6494_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
= X ) ).
% scomp_Pair
thf(fact_6495_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F3: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F3 )
= ( F3 @ X ) ) ).
% Pair_scomp
thf(fact_6496_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_6497_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_6498_prod__list__def,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A )
= ( groups_monoid_F @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod_list_def
thf(fact_6499_Zfun__imp__Zfun,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F5: filter @ A,G3: A > C,K5: real] :
( ( zfun @ A @ B @ F3 @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G3 @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) ) @ K5 ) )
@ F5 )
=> ( zfun @ A @ C @ G3 @ F5 ) ) ) ) ).
% Zfun_imp_Zfun
thf(fact_6500_Zfun__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F5: filter @ A] :
( zfun @ A @ B
@ ^ [X4: A] : ( zero_zero @ B )
@ F5 ) ) ).
% Zfun_zero
thf(fact_6501_Zfun__mult,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F5: filter @ D,G3: D > A] :
( ( zfun @ D @ A @ F3 @ F5 )
=> ( ( zfun @ D @ A @ G3 @ F5 )
=> ( zfun @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ F5 ) ) ) ) ).
% Zfun_mult
thf(fact_6502_Zfun__mult__left,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F5: filter @ D,A3: A] :
( ( zfun @ D @ A @ F3 @ F5 )
=> ( zfun @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ A3 )
@ F5 ) ) ) ).
% Zfun_mult_left
thf(fact_6503_Zfun__mult__right,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F5: filter @ D,A3: A] :
( ( zfun @ D @ A @ F3 @ F5 )
=> ( zfun @ D @ A
@ ^ [X4: D] : ( times_times @ A @ A3 @ ( F3 @ X4 ) )
@ F5 ) ) ) ).
% Zfun_mult_right
thf(fact_6504_sum__list__def,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( groups_monoid_F @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_def
thf(fact_6505_decseq__ge,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X8: nat > A,L5: A,N: nat] :
( ( order_antimono @ nat @ A @ X8 )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ L5 @ ( X8 @ N ) ) ) ) ) ).
% decseq_ge
thf(fact_6506_not__in__connected__cases,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A,X: A] :
( ( topolo1966860045006549960nected @ A @ S3 )
=> ( ~ ( member @ A @ X @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( condit941137186595557371_above @ A @ S3 )
=> ~ ! [Y6: A] :
( ( member @ A @ Y6 @ S3 )
=> ( ord_less_eq @ A @ Y6 @ X ) ) )
=> ~ ( ( condit1013018076250108175_below @ A @ S3 )
=> ~ ! [Y6: A] :
( ( member @ A @ Y6 @ S3 )
=> ( ord_less_eq @ A @ X @ Y6 ) ) ) ) ) ) ) ) ).
% not_in_connected_cases
thf(fact_6507_decseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,I2: nat,J2: nat] :
( ( order_antimono @ nat @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ A @ ( F3 @ J2 ) @ ( F3 @ I2 ) ) ) ) ) ).
% decseqD
thf(fact_6508_decseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X7: nat > A] :
! [M6: nat,N5: nat] :
( ( ord_less_eq @ nat @ M6 @ N5 )
=> ( ord_less_eq @ A @ ( X7 @ N5 ) @ ( X7 @ M6 ) ) ) ) ) ) ).
% decseq_def
thf(fact_6509_connected__iff__interval,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ( ( topolo1966860045006549960nected @ A )
= ( ^ [U3: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ U3 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ U3 )
=> ! [Z4: A] :
( ( ord_less_eq @ A @ X4 @ Z4 )
=> ( ( ord_less_eq @ A @ Z4 @ Y4 )
=> ( member @ A @ Z4 @ U3 ) ) ) ) ) ) ) ) ).
% connected_iff_interval
thf(fact_6510_connectedI__interval,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [U4: set @ A] :
( ! [X3: A,Y3: A,Z: A] :
( ( member @ A @ X3 @ U4 )
=> ( ( member @ A @ Y3 @ U4 )
=> ( ( ord_less_eq @ A @ X3 @ Z )
=> ( ( ord_less_eq @ A @ Z @ Y3 )
=> ( member @ A @ Z @ U4 ) ) ) ) )
=> ( topolo1966860045006549960nected @ A @ U4 ) ) ) ).
% connectedI_interval
thf(fact_6511_connectedD__interval,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [U4: set @ A,X: A,Y: A,Z3: A] :
( ( topolo1966860045006549960nected @ A @ U4 )
=> ( ( member @ A @ X @ U4 )
=> ( ( member @ A @ Y @ U4 )
=> ( ( ord_less_eq @ A @ X @ Z3 )
=> ( ( ord_less_eq @ A @ Z3 @ Y )
=> ( member @ A @ Z3 @ U4 ) ) ) ) ) ) ) ).
% connectedD_interval
thf(fact_6512_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F4: A > B] :
! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F4 @ Y4 ) @ ( F4 @ X4 ) ) ) ) ) ) ).
% antimono_def
thf(fact_6513_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ Y3 ) @ ( F3 @ X3 ) ) )
=> ( order_antimono @ A @ B @ F3 ) ) ) ).
% antimonoI
thf(fact_6514_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) ) ) ) ).
% antimonoE
thf(fact_6515_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) ) ) ) ).
% antimonoD
thf(fact_6516_decseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: nat > A,I2: nat] :
( ( order_antimono @ nat @ A @ A6 )
=> ( ord_less_eq @ A @ ( A6 @ ( suc @ I2 ) ) @ ( A6 @ I2 ) ) ) ) ).
% decseq_SucD
thf(fact_6517_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
=> ( order_antimono @ nat @ A @ X8 ) ) ) ).
% decseq_SucI
thf(fact_6518_decseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N5: nat] : ( ord_less_eq @ A @ ( F4 @ ( suc @ N5 ) ) @ ( F4 @ N5 ) ) ) ) ) ).
% decseq_Suc_iff
thf(fact_6519_antimono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_antimono @ nat @ A
@ ^ [I: nat] : ( compow @ ( A > A ) @ I @ Q @ ( top_top @ A ) ) ) ) ) ).
% antimono_funpow
thf(fact_6520_independent__span__bound,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [T6: set @ A,S3: set @ A] :
( ( finite_finite2 @ A @ T6 )
=> ( ~ ( real_V358717886546972837endent @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( real_Vector_span @ A @ T6 ) )
=> ( ( finite_finite2 @ A @ S3 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ S3 ) @ ( finite_card @ A @ T6 ) ) ) ) ) ) ) ).
% independent_span_bound
thf(fact_6521_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_6522_span__insert__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( real_Vector_span @ A @ ( insert @ A @ ( zero_zero @ A ) @ S3 ) )
= ( real_Vector_span @ A @ S3 ) ) ) ).
% span_insert_0
thf(fact_6523_span__empty,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% span_empty
thf(fact_6524_span__delete__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_Vector_span @ A @ S3 ) ) ) ).
% span_delete_0
thf(fact_6525_span__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] : ( member @ A @ ( zero_zero @ A ) @ ( real_Vector_span @ A @ S3 ) ) ) ).
% span_0
thf(fact_6526_span__induct__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,S3: set @ A,H: A > $o] :
( ( member @ A @ X @ ( real_Vector_span @ A @ S3 ) )
=> ( ( H @ ( zero_zero @ A ) )
=> ( ! [C2: real,X3: A,Y3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( H @ Y3 )
=> ( H @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ X3 ) @ Y3 ) ) ) )
=> ( H @ X ) ) ) ) ) ).
% span_induct_alt
thf(fact_6527_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_6528_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,F3: ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ A @ B ) ),P2: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ 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 @ P2 ) ) ) ) )
=> ( P2 @ A5 @ B4 ) )
=> ( P2 @ A3 @ B2 ) ) ) ) ).
% lfp_induct2
thf(fact_6529_lfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,P2: A > $o] :
( ( order_mono @ A @ A @ F3 )
=> ( ! [S4: A] :
( ( P2 @ S4 )
=> ( ( ord_less_eq @ A @ S4 @ ( complete_lattice_lfp @ A @ F3 ) )
=> ( P2 @ ( F3 @ S4 ) ) ) )
=> ( ! [M8: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ M8 )
=> ( P2 @ X5 ) )
=> ( P2 @ ( complete_Sup_Sup @ A @ M8 ) ) )
=> ( P2 @ ( complete_lattice_lfp @ A @ F3 ) ) ) ) ) ) ).
% lfp_ordinal_induct
thf(fact_6530_def__lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: A,F3: A > A,P2: A] :
( ( A6
= ( complete_lattice_lfp @ A @ F3 ) )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ ( inf_inf @ A @ A6 @ P2 ) ) @ P2 )
=> ( ord_less_eq @ A @ A6 @ P2 ) ) ) ) ) ).
% def_lfp_induct
thf(fact_6531_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X7: $o > A,Y10: $o > A] :
( ( ord_less_eq @ A @ ( X7 @ $false ) @ ( Y10 @ $false ) )
& ( ord_less_eq @ A @ ( X7 @ $true ) @ ( Y10 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_6532_lfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,G3: A > A] :
( ! [Z9: A] : ( ord_less_eq @ A @ ( F3 @ Z9 ) @ ( G3 @ Z9 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ ( complete_lattice_lfp @ A @ G3 ) ) ) ) ).
% lfp_mono
thf(fact_6533_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_6534_lfp__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,A6: A] :
( ! [U5: A] :
( ( ord_less_eq @ A @ ( F3 @ U5 ) @ U5 )
=> ( ord_less_eq @ A @ A6 @ U5 ) )
=> ( ord_less_eq @ A @ A6 @ ( complete_lattice_lfp @ A @ F3 ) ) ) ) ).
% lfp_greatest
thf(fact_6535_lfp__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A > A] :
( ! [X3: A,Y3: A,W: A,Z: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less_eq @ A @ ( F3 @ X3 @ W ) @ ( F3 @ Y3 @ Z ) ) ) )
=> ( ( 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_6536_lfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F5: A > A,X: A] :
( ( order_mono @ A @ A @ F5 )
=> ( ( ( F5 @ X )
= X )
=> ( ! [Z: A] :
( ( ( F5 @ Z )
= Z )
=> ( ord_less_eq @ A @ X @ Z ) )
=> ( ( complete_lattice_lfp @ A @ F5 )
= X ) ) ) ) ) ).
% lfp_eqI
thf(fact_6537_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_6538_lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,P2: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ ( inf_inf @ A @ ( complete_lattice_lfp @ A @ F3 ) @ P2 ) ) @ P2 )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ P2 ) ) ) ) ).
% lfp_induct
thf(fact_6539_lfp__transfer__bounded,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [P2: A > $o,F3: A > A,Alpha: A > B,G3: B > B] :
( ( P2 @ ( bot_bot @ A ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( P2 @ ( F3 @ X3 ) ) )
=> ( ! [M8: nat > A] :
( ! [I4: nat] : ( P2 @ ( M8 @ I4 ) )
=> ( P2 @ ( complete_Sup_Sup @ A @ ( image2 @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) ) )
=> ( ! [M8: nat > A] :
( ( order_mono @ nat @ A @ M8 )
=> ( ! [I4: nat] : ( P2 @ ( M8 @ I4 ) )
=> ( ( Alpha @ ( complete_Sup_Sup @ A @ ( image2 @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ nat @ B
@ ^ [I: nat] : ( Alpha @ ( M8 @ I ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ( order_sup_continuous @ A @ A @ F3 )
=> ( ( order_sup_continuous @ B @ B @ G3 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ ( complete_lattice_lfp @ A @ F3 ) )
=> ( ( Alpha @ ( F3 @ X3 ) )
= ( G3 @ ( Alpha @ X3 ) ) ) ) )
=> ( ! [X3: B] : ( ord_less_eq @ B @ ( Alpha @ ( bot_bot @ A ) ) @ ( G3 @ X3 ) )
=> ( ( Alpha @ ( complete_lattice_lfp @ A @ F3 ) )
= ( complete_lattice_lfp @ B @ G3 ) ) ) ) ) ) ) ) ) ) ) ).
% lfp_transfer_bounded
thf(fact_6540_dim__le__card,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [V: set @ A,W4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ W4 ) )
=> ( ( finite_finite2 @ A @ W4 )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ V ) @ ( finite_card @ A @ W4 ) ) ) ) ) ).
% dim_le_card
thf(fact_6541_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 )
=> ( ! [X3: B] : ( ord_less_eq @ B @ ( Alpha @ ( bot_bot @ A ) ) @ ( G3 @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less_eq @ A @ X3 @ ( complete_lattice_lfp @ A @ F3 ) )
=> ( ( Alpha @ ( F3 @ X3 ) )
= ( G3 @ ( Alpha @ X3 ) ) ) )
=> ( ( Alpha @ ( complete_lattice_lfp @ A @ F3 ) )
= ( complete_lattice_lfp @ B @ G3 ) ) ) ) ) ) ) ) ).
% lfp_transfer
thf(fact_6542_dim__le__card_H,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ S2 ) @ ( finite_card @ A @ S2 ) ) ) ) ).
% dim_le_card'
thf(fact_6543_span__card__ge__dim,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B6: set @ A,V: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ V )
=> ( ( ord_less_eq @ ( set @ A ) @ V @ ( real_Vector_span @ A @ B6 ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ord_less_eq @ nat @ ( real_Vector_dim @ A @ V ) @ ( finite_card @ A @ B6 ) ) ) ) ) ) ).
% span_card_ge_dim
thf(fact_6544_dim__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_dim @ A )
= ( ^ [V5: set @ A] :
( if @ nat
@ ? [B5: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ B5 )
& ( ( real_Vector_span @ A @ B5 )
= ( real_Vector_span @ A @ V5 ) ) )
@ ( finite_card @ A
@ ( fChoice @ ( set @ A )
@ ^ [B5: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ B5 )
& ( ( real_Vector_span @ A @ B5 )
= ( real_Vector_span @ A @ V5 ) ) ) ) )
@ ( zero_zero @ nat ) ) ) ) ) ).
% dim_def
thf(fact_6545_linear__indep__image__lemma,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F3: A > B,B6: set @ A,X: A] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ~ ( real_V358717886546972837endent @ B @ ( image2 @ A @ B @ F3 @ B6 ) )
=> ( ( inj_on @ A @ B @ F3 @ B6 )
=> ( ( member @ A @ X @ ( real_Vector_span @ A @ B6 ) )
=> ( ( ( F3 @ X )
= ( zero_zero @ B ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ) ) ) ) ).
% linear_indep_image_lemma
thf(fact_6546_some__equality,axiom,
! [A: $tType,P2: A > $o,A3: A] :
( ( P2 @ A3 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( X3 = A3 ) )
=> ( ( fChoice @ A @ P2 )
= A3 ) ) ) ).
% some_equality
thf(fact_6547_some__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ^ [Y4: A] : Y4 = X )
= X ) ).
% some_eq_trivial
thf(fact_6548_some__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X ) )
= X ) ).
% some_sym_eq_trivial
thf(fact_6549_linear__eq__0__on__span,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F3: A > B,B2: set @ A,X: A] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( ( F3 @ X3 )
= ( zero_zero @ B ) ) )
=> ( ( member @ A @ X @ ( real_Vector_span @ A @ B2 ) )
=> ( ( F3 @ X )
= ( zero_zero @ B ) ) ) ) ) ) ).
% linear_eq_0_on_span
thf(fact_6550_some__in__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
@ A6 )
= ( A6
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_6551_someI,axiom,
! [A: $tType,P2: A > $o,X: A] :
( ( P2 @ X )
=> ( P2 @ ( fChoice @ A @ P2 ) ) ) ).
% someI
thf(fact_6552_Eps__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( fChoice @ A @ P2 )
= ( fChoice @ A @ Q ) ) ) ).
% Eps_cong
thf(fact_6553_tfl__some,axiom,
! [A: $tType,P9: A > $o,X5: A] :
( ( P9 @ X5 )
=> ( P9 @ ( fChoice @ A @ P9 ) ) ) ).
% tfl_some
thf(fact_6554_some__eq__imp,axiom,
! [A: $tType,P2: A > $o,A3: A,B2: A] :
( ( ( fChoice @ A @ P2 )
= A3 )
=> ( ( P2 @ B2 )
=> ( P2 @ A3 ) ) ) ).
% some_eq_imp
thf(fact_6555_someI2,axiom,
! [A: $tType,P2: A > $o,A3: A,Q: A > $o] :
( ( P2 @ A3 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice @ A @ P2 ) ) ) ) ).
% someI2
thf(fact_6556_someI__ex,axiom,
! [A: $tType,P2: A > $o] :
( ? [X_1: A] : ( P2 @ X_1 )
=> ( P2 @ ( fChoice @ A @ P2 ) ) ) ).
% someI_ex
thf(fact_6557_someI2__ex,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P2 @ X_1 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice @ A @ P2 ) ) ) ) ).
% someI2_ex
thf(fact_6558_someI2__bex,axiom,
! [A: $tType,A6: set @ A,P2: A > $o,Q: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ( P2 @ X5 ) )
=> ( ! [X3: A] :
( ( ( member @ A @ X3 @ A6 )
& ( P2 @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_6559_some__eq__ex,axiom,
! [A: $tType,P2: A > $o] :
( ( P2 @ ( fChoice @ A @ P2 ) )
= ( ? [X7: A] : ( P2 @ X7 ) ) ) ).
% some_eq_ex
thf(fact_6560_some1__equality,axiom,
! [A: $tType,P2: A > $o,A3: A] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y3: A] :
( ( P2 @ Y3 )
=> ( Y3 = X5 ) ) )
=> ( ( P2 @ A3 )
=> ( ( fChoice @ A @ P2 )
= A3 ) ) ) ).
% some1_equality
thf(fact_6561_linear__times__of__real,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A3: A] :
( real_Vector_linear @ real @ A
@ ^ [X4: real] : ( times_times @ A @ A3 @ ( real_Vector_of_real @ A @ X4 ) ) ) ) ).
% linear_times_of_real
thf(fact_6562_linear__scale__real,axiom,
! [F3: real > real,R3: real,B2: real] :
( ( real_Vector_linear @ real @ real @ F3 )
=> ( ( F3 @ ( times_times @ real @ R3 @ B2 ) )
= ( times_times @ real @ R3 @ ( F3 @ B2 ) ) ) ) ).
% linear_scale_real
thf(fact_6563_real__linearD,axiom,
! [F3: real > real] :
( ( real_Vector_linear @ real @ real @ F3 )
=> ~ ! [C2: real] :
( F3
!= ( times_times @ real @ C2 ) ) ) ).
% real_linearD
thf(fact_6564_linear__times,axiom,
! [A: $tType] :
( ( real_V6157519004096292374lgebra @ A )
=> ! [C3: A] : ( real_Vector_linear @ A @ A @ ( times_times @ A @ C3 ) ) ) ).
% linear_times
thf(fact_6565_linear__0,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F3: A > B] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( F3 @ ( zero_zero @ A ) )
= ( zero_zero @ B ) ) ) ) ).
% linear_0
thf(fact_6566_module__hom__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ( real_Vector_linear @ A @ B
@ ^ [X4: A] : ( zero_zero @ B ) ) ) ).
% module_hom_zero
thf(fact_6567_exE__some,axiom,
! [A: $tType,P2: A > $o,C3: A] :
( ? [X_1: A] : ( P2 @ X_1 )
=> ( ( C3
= ( fChoice @ A @ P2 ) )
=> ( P2 @ C3 ) ) ) ).
% exE_some
thf(fact_6568_linear__injective__0,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F3: A > B] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( ( F3 @ X4 )
= ( zero_zero @ B ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% linear_injective_0
thf(fact_6569_representation__scale,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V3: A,R3: real] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V3 @ ( real_Vector_span @ A @ Basis ) )
=> ( ( real_V7696804695334737415tation @ A @ Basis @ ( real_V8093663219630862766scaleR @ A @ R3 @ V3 ) )
= ( ^ [B5: A] : ( times_times @ real @ R3 @ ( real_V7696804695334737415tation @ A @ Basis @ V3 @ B5 ) ) ) ) ) ) ) ).
% representation_scale
thf(fact_6570_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_6571_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y8: B] :
( ( X = X9 )
& ( Y = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% Eps_case_prod_eq
thf(fact_6572_gfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F5: A > A,X: A] :
( ( order_mono @ A @ A @ F5 )
=> ( ( ( F5 @ X )
= X )
=> ( ! [Z: A] :
( ( ( F5 @ Z )
= Z )
=> ( ord_less_eq @ A @ Z @ X ) )
=> ( ( complete_lattice_gfp @ A @ F5 )
= X ) ) ) ) ) ).
% gfp_eqI
thf(fact_6573_gfp__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A > A] :
( ! [X3: A,Y3: A,W: A,Z: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less_eq @ A @ ( F3 @ X3 @ W ) @ ( F3 @ Y3 @ Z ) ) ) )
=> ( ( 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_6574_gfp__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,X8: A] :
( ! [U5: A] :
( ( ord_less_eq @ A @ U5 @ ( F3 @ U5 ) )
=> ( ord_less_eq @ A @ U5 @ X8 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F3 ) @ X8 ) ) ) ).
% gfp_least
thf(fact_6575_gfp__upperbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X8: A,F3: A > A] :
( ( ord_less_eq @ A @ X8 @ ( F3 @ X8 ) )
=> ( ord_less_eq @ A @ X8 @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ).
% gfp_upperbound
thf(fact_6576_gfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,G3: A > A] :
( ! [Z9: A] : ( ord_less_eq @ A @ ( F3 @ Z9 ) @ ( G3 @ Z9 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F3 ) @ ( complete_lattice_gfp @ A @ G3 ) ) ) ) ).
% gfp_mono
thf(fact_6577_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P4: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] : ( P4 @ ( product_Pair @ A @ B @ A7 @ B5 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_6578_representation__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A] :
( ( real_V7696804695334737415tation @ A @ Basis @ ( zero_zero @ A ) )
= ( ^ [B5: A] : ( zero_zero @ real ) ) ) ) ).
% representation_zero
thf(fact_6579_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_6580_coinduct__lemma,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X8: A,F3: A > A] :
( ( ord_less_eq @ A @ X8 @ ( F3 @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F3 ) ) ) )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F3 ) ) @ ( F3 @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_6581_def__coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: A,F3: A > A,X8: A] :
( ( A6
= ( complete_lattice_gfp @ A @ F3 ) )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ X8 @ ( F3 @ ( sup_sup @ A @ X8 @ A6 ) ) )
=> ( ord_less_eq @ A @ X8 @ A6 ) ) ) ) ) ).
% def_coinduct
thf(fact_6582_coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,X8: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ X8 @ ( F3 @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F3 ) ) ) )
=> ( ord_less_eq @ A @ X8 @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ) ).
% coinduct
thf(fact_6583_gfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,P2: A > $o] :
( ( order_mono @ A @ A @ F3 )
=> ( ! [S4: A] :
( ( P2 @ S4 )
=> ( ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F3 ) @ S4 )
=> ( P2 @ ( F3 @ S4 ) ) ) )
=> ( ! [M8: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ M8 )
=> ( P2 @ X5 ) )
=> ( P2 @ ( complete_Inf_Inf @ A @ M8 ) ) )
=> ( P2 @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ) ) ).
% gfp_ordinal_induct
thf(fact_6584_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_6585_Eps__case__prod,axiom,
! [B: $tType,A: $tType,P2: A > B > $o] :
( ( fChoice @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) )
= ( fChoice @ ( product_prod @ A @ B )
@ ^ [Xy2: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Xy2 ) @ ( product_snd @ A @ B @ Xy2 ) ) ) ) ).
% Eps_case_prod
thf(fact_6586_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_6587_gfp__transfer__bounded,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [P2: A > $o,F3: A > A,Alpha: A > B,G3: B > B] :
( ( P2 @ ( F3 @ ( top_top @ A ) ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( P2 @ ( F3 @ X3 ) ) )
=> ( ! [M8: nat > A] :
( ( order_antimono @ nat @ A @ M8 )
=> ( ! [I4: nat] : ( P2 @ ( M8 @ I4 ) )
=> ( P2 @ ( complete_Inf_Inf @ A @ ( image2 @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ! [M8: nat > A] :
( ( order_antimono @ nat @ A @ M8 )
=> ( ! [I4: nat] : ( P2 @ ( M8 @ I4 ) )
=> ( ( Alpha @ ( complete_Inf_Inf @ A @ ( image2 @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ nat @ B
@ ^ [I: nat] : ( Alpha @ ( M8 @ I ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ( order_inf_continuous @ A @ A @ F3 )
=> ( ( order_inf_continuous @ B @ B @ G3 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( ( Alpha @ ( F3 @ X3 ) )
= ( G3 @ ( Alpha @ X3 ) ) ) )
=> ( ! [X3: B] : ( ord_less_eq @ B @ ( G3 @ X3 ) @ ( Alpha @ ( F3 @ ( top_top @ A ) ) ) )
=> ( ( Alpha @ ( complete_lattice_gfp @ A @ F3 ) )
= ( complete_lattice_gfp @ B @ G3 ) ) ) ) ) ) ) ) ) ) ) ).
% gfp_transfer_bounded
thf(fact_6588_list__ex__length,axiom,
! [A: $tType] :
( ( list_ex @ A )
= ( ^ [P4: A > $o,Xs: list @ A] :
? [N5: nat] :
( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P4 @ ( nth @ A @ Xs @ N5 ) ) ) ) ) ).
% list_ex_length
thf(fact_6589_max__ext_Ocases,axiom,
! [A: $tType,A12: set @ A,A23: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A12 @ A23 ) @ ( max_ext @ A @ R2 ) )
=> ~ ( ( finite_finite2 @ A @ A12 )
=> ( ( finite_finite2 @ A @ A23 )
=> ( ( A23
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A12 )
=> ? [Xa4: A] :
( ( member @ A @ Xa4 @ A23 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa4 ) @ R2 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_6590_max__ext_Osimps,axiom,
! [A: $tType,A12: set @ A,A23: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A12 @ A23 ) @ ( max_ext @ A @ R2 ) )
= ( ( finite_finite2 @ A @ A12 )
& ( finite_finite2 @ A @ A23 )
& ( A23
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A12 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A23 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% max_ext.simps
thf(fact_6591_max__ext_Omax__extI,axiom,
! [A: $tType,X8: set @ A,Y7: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X8 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( Y7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Y7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa ) @ R2 ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X8 @ Y7 ) @ ( max_ext @ A @ R2 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_6592_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,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R6 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_6593_max__extp__max__ext__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) )
= ( ^ [X4: set @ A,Y4: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y4 ) @ ( max_ext @ A @ R2 ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_6594_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
@ ^ [X7: set @ A,Y10: set @ A] :
( ( finite_finite2 @ A @ X7 )
& ( finite_finite2 @ A @ Y10 )
& ( Y10
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ Y10 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R6 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_6595_finite__sequence__to__countable__set,axiom,
! [A: $tType,X8: set @ A] :
( ( countable_countable @ A @ X8 )
=> ~ ! [F10: nat > ( set @ A )] :
( ! [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( F10 @ I4 ) @ X8 )
=> ( ! [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( F10 @ I4 ) @ ( F10 @ ( suc @ I4 ) ) )
=> ( ! [I4: nat] : ( finite_finite2 @ A @ ( F10 @ I4 ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F10 @ ( top_top @ ( set @ nat ) ) ) )
!= X8 ) ) ) ) ) ).
% finite_sequence_to_countable_set
thf(fact_6596_ccInf__greatest,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,Z3: A] :
( ( countable_countable @ A @ A6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ).
% ccInf_greatest
thf(fact_6597_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_6598_ccInf__lower2,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,U: A,V3: A] :
( ( countable_countable @ A @ A6 )
=> ( ( member @ A @ U @ A6 )
=> ( ( ord_less_eq @ A @ U @ V3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ V3 ) ) ) ) ) ).
% ccInf_lower2
thf(fact_6599_ccInf__lower,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,X: A] :
( ( countable_countable @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ X ) ) ) ) ).
% ccInf_lower
thf(fact_6600_ccInf__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B6: set @ A,A6: set @ A] :
( ( countable_countable @ A @ B6 )
=> ( ( countable_countable @ A @ A6 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ( ord_less_eq @ A @ X5 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ) ) ).
% ccInf_mono
thf(fact_6601_ccSup__upper2,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,U: A,V3: A] :
( ( countable_countable @ A @ A6 )
=> ( ( member @ A @ U @ A6 )
=> ( ( ord_less_eq @ A @ V3 @ U )
=> ( ord_less_eq @ A @ V3 @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ).
% ccSup_upper2
thf(fact_6602_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_6603_ccSup__upper,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,X: A] :
( ( countable_countable @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% ccSup_upper
thf(fact_6604_ccSup__least,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,Z3: A] :
( ( countable_countable @ A @ A6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ Z3 ) ) ) ) ).
% ccSup_least
thf(fact_6605_ccSup__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B6: set @ A,A6: set @ A] :
( ( countable_countable @ A @ B6 )
=> ( ( countable_countable @ A @ A6 )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ B6 )
& ( ord_less_eq @ A @ A5 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ).
% ccSup_mono
thf(fact_6606_ccSup__subset__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B6: set @ A,A6: set @ A] :
( ( countable_countable @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ).
% ccSup_subset_mono
thf(fact_6607_ccInf__superset__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ( countable_countable @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ) ).
% ccInf_superset_mono
thf(fact_6608_ccSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,I2: B,U: A,F3: B > A] :
( ( countable_countable @ B @ A6 )
=> ( ( member @ B @ I2 @ A6 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ I2 ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ) ).
% ccSUP_upper2
thf(fact_6609_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_6610_ccSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,I2: B,F3: B > A] :
( ( countable_countable @ B @ A6 )
=> ( ( member @ B @ I2 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) ) ) ) ) ).
% ccSUP_upper
thf(fact_6611_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_6612_ccSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,B6: set @ C,F3: B > A,G3: C > A] :
( ( countable_countable @ B @ A6 )
=> ( ( countable_countable @ C @ B6 )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B6 )
& ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G3 @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% ccSUP_mono
thf(fact_6613_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_6614_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_6615_ccINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,I2: B,F3: B > A,U: A] :
( ( countable_countable @ B @ A6 )
=> ( ( member @ B @ I2 @ A6 )
=> ( ( ord_less_eq @ A @ ( F3 @ I2 ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ U ) ) ) ) ) ).
% ccINF_lower2
thf(fact_6616_ccINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,I2: B,F3: B > A] :
( ( countable_countable @ B @ A6 )
=> ( ( member @ B @ I2 @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( F3 @ I2 ) ) ) ) ) ).
% ccINF_lower
thf(fact_6617_ccINF__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,B6: set @ C,F3: B > A,G3: C > A] :
( ( countable_countable @ B @ A6 )
=> ( ( countable_countable @ C @ B6 )
=> ( ! [M: C] :
( ( member @ C @ M @ B6 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A6 )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ M ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% ccINF_mono
thf(fact_6618_ccSup__inter__less__eq,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ( countable_countable @ A @ A6 )
=> ( ( countable_countable @ A @ B6 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ).
% ccSup_inter_less_eq
thf(fact_6619_less__eq__ccInf__inter,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ A,B6: set @ A] :
( ( countable_countable @ A @ A6 )
=> ( ( countable_countable @ A @ B6 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B6 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) ) ) ) ) ).
% less_eq_ccInf_inter
thf(fact_6620_ccSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B6: set @ B,A6: set @ B,F3: B > A,G3: B > A] :
( ( countable_countable @ B @ B6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% ccSUP_subset_mono
thf(fact_6621_ccINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A6: set @ B,B6: set @ B,F3: B > A,G3: B > A] :
( ( countable_countable @ B @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ A6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B6 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% ccINF_superset_mono
thf(fact_6622_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_6623_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_6624_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_6625_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_6626_map__project__def,axiom,
! [B: $tType,A: $tType] :
( ( map_project @ A @ B )
= ( ^ [F4: A > ( option @ B ),A8: set @ A] :
( collect @ B
@ ^ [B5: B] :
? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ( ( F4 @ X4 )
= ( some @ B @ B5 ) ) ) ) ) ) ).
% map_project_def
thf(fact_6627_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ^ [Uu3: product_prod @ ( set @ A ) @ ( set @ A )] :
? [X7: set @ A,Y10: set @ A] :
( ( Uu3
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X7 @ Y10 ) )
& ( X7
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ Y10 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ X7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_6628_Chains__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
( collect @ ( set @ A )
@ ^ [C7: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ C7 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ C7 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% Chains_def
thf(fact_6629_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_6630_Chains__subset_H,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) ) )
@ ( chains @ A @ R3 ) ) ) ).
% Chains_subset'
thf(fact_6631_Chains__alt__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( ( chains @ A @ R3 )
= ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) ) ) ) ) ).
% Chains_alt_def
thf(fact_6632_Chains__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( set @ A ) ) @ ( chains @ A @ R3 )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) ) ) ) ).
% Chains_subset
thf(fact_6633_Zorns__po__lemma,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ! [C8: set @ A] :
( ( member @ ( set @ A ) @ C8 @ ( chains @ A @ R3 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R3 ) )
& ! [Xa4: A] :
( ( member @ A @ Xa4 @ C8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa4 @ X5 ) @ R3 ) ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa ) @ R3 )
=> ( Xa = X3 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_6634_construct__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ( ( real_V4425403222259421789struct @ A @ B )
= ( ^ [B8: set @ A,G4: A > B,V4: A] :
( groups7311177749621191930dd_sum @ A @ B
@ ^ [B5: A] : ( real_V8093663219630862766scaleR @ B @ ( real_V7696804695334737415tation @ A @ ( real_V4986007116245087402_basis @ A @ B8 ) @ V4 @ B5 ) @ ( if @ B @ ( member @ A @ B5 @ B8 ) @ ( G4 @ B5 ) @ ( zero_zero @ B ) ) )
@ ( collect @ A
@ ^ [B5: A] :
( ( real_V7696804695334737415tation @ A @ ( real_V4986007116245087402_basis @ A @ B8 ) @ V4 @ B5 )
!= ( zero_zero @ real ) ) ) ) ) ) ) ).
% construct_def
thf(fact_6635_construct__outside,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [B6: set @ A,V3: A,F3: A > B] :
( ~ ( real_V358717886546972837endent @ A @ B6 )
=> ( ( member @ A @ V3 @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ ( real_V4986007116245087402_basis @ A @ B6 ) @ B6 ) ) )
=> ( ( real_V4425403222259421789struct @ A @ B @ B6 @ F3 @ V3 )
= ( zero_zero @ B ) ) ) ) ) ).
% construct_outside
thf(fact_6636_natLeq__Partial__order,axiom,
order_7125193373082350890der_on @ nat @ ( field2 @ nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).
% natLeq_Partial_order
thf(fact_6637_natLeq__Preorder,axiom,
order_preorder_on @ nat @ ( field2 @ nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).
% natLeq_Preorder
thf(fact_6638_Range__insert,axiom,
! [A: $tType,B: $tType,A3: B,B2: A,R3: set @ ( product_prod @ B @ A )] :
( ( range2 @ B @ A @ ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R3 ) )
= ( insert @ A @ B2 @ ( range2 @ B @ A @ R3 ) ) ) ).
% Range_insert
thf(fact_6639_Range_Ocases,axiom,
! [B: $tType,A: $tType,A3: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range2 @ A @ B @ R3 ) )
=> ~ ! [A5: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ A3 ) @ R3 ) ) ).
% Range.cases
thf(fact_6640_Range_Osimps,axiom,
! [B: $tType,A: $tType,A3: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range2 @ A @ B @ R3 ) )
= ( ? [A7: A,B5: B] :
( ( A3 = B5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R3 ) ) ) ) ).
% Range.simps
thf(fact_6641_Range_Ointros,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( member @ B @ B2 @ ( range2 @ A @ B @ R3 ) ) ) ).
% Range.intros
thf(fact_6642_RangeE,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A )] :
( ( member @ A @ B2 @ ( range2 @ B @ A @ R3 ) )
=> ~ ! [A5: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A5 @ B2 ) @ R3 ) ) ).
% RangeE
thf(fact_6643_Range__iff,axiom,
! [A: $tType,B: $tType,A3: A,R3: set @ ( product_prod @ B @ A )] :
( ( member @ A @ A3 @ ( range2 @ B @ A @ R3 ) )
= ( ? [Y4: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y4 @ A3 ) @ R3 ) ) ) ).
% Range_iff
thf(fact_6644_subset__Image1__Image1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_6645_Rangep__Range__eq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( rangep @ A @ B
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R3 ) )
= ( ^ [X4: B] : ( member @ B @ X4 @ ( range2 @ A @ B @ R3 ) ) ) ) ).
% Rangep_Range_eq
thf(fact_6646_ImageI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B ),A6: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( ( member @ A @ A3 @ A6 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R3 @ A6 ) ) ) ) ).
% ImageI
thf(fact_6647_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A3: B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_6648_ImageE,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A6: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ A6 ) )
=> ~ ! [X3: B] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B2 ) @ R3 )
=> ~ ( member @ B @ X3 @ A6 ) ) ) ).
% ImageE
thf(fact_6649_Image__iff,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A6: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ A6 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ B2 ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_6650_rev__ImageI,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ A6 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R3 @ A6 ) ) ) ) ).
% rev_ImageI
thf(fact_6651_Image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ B ),S6: set @ A] :
( collect @ B
@ ^ [Y4: B] :
? [X4: A] :
( ( member @ A @ X4 @ S6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R ) ) ) ) ) ).
% Image_def
thf(fact_6652_Image__singleton,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ B @ A ),A3: B] :
( ( image @ B @ A @ R3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B5 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_6653_subset__Image__Image__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A6: set @ A,B6: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ A6 ) @ ( image @ A @ A @ R3 @ B6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ B6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R3 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_6654_Range__def,axiom,
! [B: $tType,A: $tType] :
( ( range2 @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ B )] :
( collect @ B
@ ( rangep @ A @ B
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R ) ) ) ) ) ).
% Range_def
thf(fact_6655_less__eq__int__def,axiom,
( ( ord_less_eq @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_6656_add_Ogroup__axioms,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( group @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A ) ) ) ).
% add.group_axioms
thf(fact_6657_group_Oleft__cancel,axiom,
! [A: $tType,F3: A > A > A,Z3: A,Inverse: A > A,A3: A,B2: A,C3: A] :
( ( group @ A @ F3 @ Z3 @ Inverse )
=> ( ( ( F3 @ A3 @ B2 )
= ( F3 @ A3 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% group.left_cancel
thf(fact_6658_group_Oleft__inverse,axiom,
! [A: $tType,F3: A > A > A,Z3: A,Inverse: A > A,A3: A] :
( ( group @ A @ F3 @ Z3 @ Inverse )
=> ( ( F3 @ ( Inverse @ A3 ) @ A3 )
= Z3 ) ) ).
% group.left_inverse
thf(fact_6659_group_Oright__cancel,axiom,
! [A: $tType,F3: A > A > A,Z3: A,Inverse: A > A,B2: A,A3: A,C3: A] :
( ( group @ A @ F3 @ Z3 @ Inverse )
=> ( ( ( F3 @ B2 @ A3 )
= ( F3 @ C3 @ A3 ) )
= ( B2 = C3 ) ) ) ).
% group.right_cancel
thf(fact_6660_group_Oright__inverse,axiom,
! [A: $tType,F3: A > A > A,Z3: A,Inverse: A > A,A3: A] :
( ( group @ A @ F3 @ Z3 @ Inverse )
=> ( ( F3 @ A3 @ ( Inverse @ A3 ) )
= Z3 ) ) ).
% group.right_inverse
thf(fact_6661_group_Oinverse__unique,axiom,
! [A: $tType,F3: A > A > A,Z3: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F3 @ Z3 @ Inverse )
=> ( ( ( F3 @ A3 @ B2 )
= Z3 )
=> ( ( Inverse @ A3 )
= B2 ) ) ) ).
% group.inverse_unique
thf(fact_6662_group_Oinverse__inverse,axiom,
! [A: $tType,F3: A > A > A,Z3: A,Inverse: A > A,A3: A] :
( ( group @ A @ F3 @ Z3 @ Inverse )
=> ( ( Inverse @ ( Inverse @ A3 ) )
= A3 ) ) ).
% group.inverse_inverse
thf(fact_6663_group_Oinverse__neutral,axiom,
! [A: $tType,F3: A > A > A,Z3: A,Inverse: A > A] :
( ( group @ A @ F3 @ Z3 @ Inverse )
=> ( ( Inverse @ Z3 )
= Z3 ) ) ).
% group.inverse_neutral
thf(fact_6664_group_Ogroup__left__neutral,axiom,
! [A: $tType,F3: A > A > A,Z3: A,Inverse: A > A,A3: A] :
( ( group @ A @ F3 @ Z3 @ Inverse )
=> ( ( F3 @ Z3 @ A3 )
= A3 ) ) ).
% group.group_left_neutral
thf(fact_6665_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F3: A > A > A,Z3: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F3 @ Z3 @ Inverse )
=> ( ( Inverse @ ( F3 @ A3 @ B2 ) )
= ( F3 @ ( Inverse @ B2 ) @ ( Inverse @ A3 ) ) ) ) ).
% group.inverse_distrib_swap
thf(fact_6666_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,V4: B,M6: A > ( option @ B )] : ( fun_upd @ A @ ( option @ B ) @ M6 @ K3 @ ( some @ B @ V4 ) ) )
@ Ps
@ M2 ) ) ).
% map_add_map_of_foldr
thf(fact_6667_wo__rel_OisMinim__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A6: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( bNF_We4791949203932849705sMinim @ A @ R3 @ A6 @ B2 )
= ( ( member @ A @ B2 @ A6 )
& ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ X4 ) @ R3 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_6668_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_6669_map__add__upd,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),G3: A > ( option @ B ),X: A,Y: B] :
( ( map_add @ A @ B @ F3 @ ( fun_upd @ A @ ( option @ B ) @ G3 @ X @ ( some @ B @ Y ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ F3 @ G3 ) @ X @ ( some @ B @ Y ) ) ) ).
% map_add_upd
thf(fact_6670_map__add__SomeD,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K2: B,X: A] :
( ( ( map_add @ B @ A @ M2 @ N @ K2 )
= ( some @ A @ X ) )
=> ( ( ( N @ K2 )
= ( some @ A @ X ) )
| ( ( ( N @ K2 )
= ( none @ A ) )
& ( ( M2 @ K2 )
= ( some @ A @ X ) ) ) ) ) ).
% map_add_SomeD
thf(fact_6671_map__add__Some__iff,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K2: B,X: A] :
( ( ( map_add @ B @ A @ M2 @ N @ K2 )
= ( some @ A @ X ) )
= ( ( ( N @ K2 )
= ( some @ A @ X ) )
| ( ( ( N @ K2 )
= ( none @ A ) )
& ( ( M2 @ K2 )
= ( some @ A @ X ) ) ) ) ) ).
% map_add_Some_iff
thf(fact_6672_map__add__def,axiom,
! [B: $tType,A: $tType] :
( ( map_add @ A @ B )
= ( ^ [M1: A > ( option @ B ),M22: A > ( option @ B ),X4: A] : ( case_option @ ( option @ B ) @ B @ ( M1 @ X4 ) @ ( some @ B ) @ ( M22 @ X4 ) ) ) ) ).
% map_add_def
thf(fact_6673_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_6674_iteratesp_Omono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A] :
( order_mono @ ( A > $o ) @ ( A > $o )
@ ^ [P5: A > $o,X4: A] :
( ? [Y4: A] :
( ( X4
= ( F3 @ Y4 ) )
& ( P5 @ Y4 ) )
| ? [M9: set @ A] :
( ( X4
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y4: A] :
( ( member @ A @ Y4 @ M9 )
=> ( P5 @ Y4 ) ) ) ) ) ) ).
% iteratesp.mono
thf(fact_6675_inv__o__cancel,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ F3 )
= ( id @ A ) ) ) ).
% inv_o_cancel
thf(fact_6676_inv__into__f__f,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,X: A] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( hilbert_inv_into @ A @ B @ A6 @ F3 @ ( F3 @ X ) )
= X ) ) ) ).
% inv_into_f_f
thf(fact_6677_inv__identity,axiom,
! [A: $tType] :
( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [A7: A] : A7 )
= ( ^ [A7: A] : A7 ) ) ).
% inv_identity
thf(fact_6678_inv__id,axiom,
! [A: $tType] :
( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) )
= ( id @ A ) ) ).
% inv_id
thf(fact_6679_inv__into__image__cancel,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,S3: set @ A] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ A6 )
=> ( ( image2 @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F3 ) @ ( image2 @ A @ B @ F3 @ S3 ) )
= S3 ) ) ) ).
% inv_into_image_cancel
thf(fact_6680_o__inv__o__cancel,axiom,
! [B: $tType,C: $tType,A: $tType,F3: A > B,G3: A > C] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ C @ A @ ( comp @ A @ C @ B @ G3 @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) ) @ F3 )
= G3 ) ) ).
% o_inv_o_cancel
thf(fact_6681_image__inv__into__cancel,axiom,
! [B: $tType,A: $tType,F3: B > A,A6: set @ B,A11: set @ A,B13: set @ A] :
( ( ( image2 @ B @ A @ F3 @ A6 )
= A11 )
=> ( ( ord_less_eq @ ( set @ A ) @ B13 @ A11 )
=> ( ( image2 @ B @ A @ F3 @ ( image2 @ A @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F3 ) @ B13 ) )
= B13 ) ) ) ).
% image_inv_into_cancel
thf(fact_6682_chain__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% chain_singleton
thf(fact_6683_ccpo__Sup__upper,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A6: set @ A,X: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_6684_ccpo__Sup__least,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A6: set @ A,Z3: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ Z3 ) ) ) ) ).
% ccpo_Sup_least
thf(fact_6685_bij__imp__bij__inv,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_imp_bij_inv
thf(fact_6686_bij__inv__eq__iff,axiom,
! [A: $tType,B: $tType,P: A > B,X: A,Y: B] :
( ( bij_betw @ A @ B @ P @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( X
= ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P @ Y ) )
= ( ( P @ X )
= Y ) ) ) ).
% bij_inv_eq_iff
thf(fact_6687_inv__inv__eq,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) )
= F3 ) ) ).
% inv_inv_eq
thf(fact_6688_bij__betw__inv__into__right,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ A,A11: set @ B,A4: B] :
( ( bij_betw @ A @ B @ F3 @ A6 @ A11 )
=> ( ( member @ B @ A4 @ A11 )
=> ( ( F3 @ ( hilbert_inv_into @ A @ B @ A6 @ F3 @ A4 ) )
= A4 ) ) ) ).
% bij_betw_inv_into_right
thf(fact_6689_bij__betw__inv__into__left,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,A11: set @ B,A3: A] :
( ( bij_betw @ A @ B @ F3 @ A6 @ A11 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ( hilbert_inv_into @ A @ B @ A6 @ F3 @ ( F3 @ A3 ) )
= A3 ) ) ) ).
% bij_betw_inv_into_left
thf(fact_6690_inv__into__inv__into__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,A11: set @ B,A3: A] :
( ( bij_betw @ A @ B @ F3 @ A6 @ A11 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ( hilbert_inv_into @ B @ A @ A11 @ ( hilbert_inv_into @ A @ B @ A6 @ F3 ) @ A3 )
= ( F3 @ A3 ) ) ) ) ).
% inv_into_inv_into_eq
thf(fact_6691_bij__betw__inv__into,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,B6: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A6 @ B6 )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F3 ) @ B6 @ A6 ) ) ).
% bij_betw_inv_into
thf(fact_6692_inv__equality,axiom,
! [A: $tType,B: $tType,G3: B > A,F3: A > B] :
( ! [X3: A] :
( ( G3 @ ( F3 @ X3 ) )
= X3 )
=> ( ! [Y3: B] :
( ( F3 @ ( G3 @ Y3 ) )
= Y3 )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
= G3 ) ) ) ).
% inv_equality
thf(fact_6693_inj__imp__inv__eq,axiom,
! [A: $tType,B: $tType,F3: A > B,G3: B > A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ! [X3: B] :
( ( F3 @ ( G3 @ X3 ) )
= X3 )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
= G3 ) ) ) ).
% inj_imp_inv_eq
thf(fact_6694_inv__f__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,X: A,Y: B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F3 @ X )
= Y )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ Y )
= X ) ) ) ).
% inv_f_eq
thf(fact_6695_inv__f__f,axiom,
! [B: $tType,A: $tType,F3: A > B,X: A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ ( F3 @ X ) )
= X ) ) ).
% inv_f_f
thf(fact_6696_inv__into__f__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,X: A,Y: B] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( ( F3 @ X )
= Y )
=> ( ( hilbert_inv_into @ A @ B @ A6 @ F3 @ Y )
= X ) ) ) ) ).
% inv_into_f_eq
thf(fact_6697_f__inv__into__f,axiom,
! [B: $tType,A: $tType,Y: A,F3: B > A,A6: set @ B] :
( ( member @ A @ Y @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( ( F3 @ ( hilbert_inv_into @ B @ A @ A6 @ F3 @ Y ) )
= Y ) ) ).
% f_inv_into_f
thf(fact_6698_inv__into__into,axiom,
! [A: $tType,B: $tType,X: A,F3: B > A,A6: set @ B] :
( ( member @ A @ X @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( member @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F3 @ X ) @ A6 ) ) ).
% inv_into_into
thf(fact_6699_inv__into__injective,axiom,
! [A: $tType,B: $tType,A6: set @ A,F3: A > B,X: B,Y: B] :
( ( ( hilbert_inv_into @ A @ B @ A6 @ F3 @ X )
= ( hilbert_inv_into @ A @ B @ A6 @ F3 @ Y ) )
=> ( ( member @ B @ X @ ( image2 @ A @ B @ F3 @ A6 ) )
=> ( ( member @ B @ Y @ ( image2 @ A @ B @ F3 @ A6 ) )
=> ( X = Y ) ) ) ) ).
% inv_into_injective
thf(fact_6700_surj__imp__inv__eq,axiom,
! [B: $tType,A: $tType,F3: B > A,G3: A > B] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X3: B] :
( ( G3 @ ( F3 @ X3 ) )
= X3 )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 )
= G3 ) ) ) ).
% surj_imp_inv_eq
thf(fact_6701_image__f__inv__f,axiom,
! [B: $tType,A: $tType,F3: B > A,A6: set @ A] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ B @ A @ F3 @ ( image2 @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 ) @ A6 ) )
= A6 ) ) ).
% image_f_inv_f
thf(fact_6702_surj__iff__all,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( F3 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 @ X4 ) )
= X4 ) ) ) ).
% surj_iff_all
thf(fact_6703_surj__f__inv__f,axiom,
! [B: $tType,A: $tType,F3: B > A,Y: A] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( F3 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 @ Y ) )
= Y ) ) ).
% surj_f_inv_f
thf(fact_6704_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 )
@ ^ [I: A,X4: D] : ( Q @ I @ ( F3 @ X4 ) ) ) ) ) ).
% mono_compose
thf(fact_6705_inv__into__def2,axiom,
! [B: $tType,A: $tType] :
( ( hilbert_inv_into @ A @ B )
= ( ^ [A8: set @ A,F4: A > B,X4: B] :
( fChoice @ A
@ ^ [Y4: A] :
( ( member @ A @ Y4 @ A8 )
& ( ( F4 @ Y4 )
= X4 ) ) ) ) ) ).
% inv_into_def2
thf(fact_6706_inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( hilbert_inv_into @ A @ B )
= ( ^ [A8: set @ A,F4: A > B,X4: B] :
( fChoice @ A
@ ^ [Y4: A] :
( ( member @ A @ Y4 @ A8 )
& ( ( F4 @ Y4 )
= X4 ) ) ) ) ) ).
% inv_into_def
thf(fact_6707_inv__def,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 )
= ( ^ [Y4: A] :
( fChoice @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
= Y4 ) ) ) ) ).
% inv_def
thf(fact_6708_inj__transfer,axiom,
! [B: $tType,A: $tType,F3: A > B,P2: A > $o,X: A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ! [Y3: B] :
( ( member @ B @ Y3 @ ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
=> ( P2 @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ Y3 ) ) )
=> ( P2 @ X ) ) ) ).
% inj_transfer
thf(fact_6709_image__inv__f__f,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( image2 @ A @ B @ F3 @ A6 ) )
= A6 ) ) ).
% image_inv_f_f
thf(fact_6710_inj__imp__surj__inv,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% inj_imp_surj_inv
thf(fact_6711_surj__imp__inj__inv,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% surj_imp_inj_inv
thf(fact_6712_inj__on__inv__into,axiom,
! [B: $tType,A: $tType,B6: set @ A,F3: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F3 @ A6 ) )
=> ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F3 ) @ B6 ) ) ).
% inj_on_inv_into
thf(fact_6713_inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > B,G3: C > A,A6: set @ C,X: B] :
( ( inj_on @ A @ B @ F3 @ ( image2 @ C @ A @ G3 @ A6 ) )
=> ( ( inj_on @ C @ A @ G3 @ A6 )
=> ( ( member @ B @ X @ ( image2 @ A @ B @ F3 @ ( image2 @ C @ A @ G3 @ A6 ) ) )
=> ( ( hilbert_inv_into @ C @ B @ A6 @ ( comp @ A @ B @ C @ F3 @ G3 ) @ X )
= ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ A6 @ G3 ) @ ( hilbert_inv_into @ A @ B @ ( image2 @ C @ A @ G3 @ A6 ) @ F3 ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_6714_bij__betw__inv__into__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,A11: set @ B,B6: set @ A,B13: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A6 @ A11 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( ( ( image2 @ A @ B @ F3 @ B6 )
= B13 )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F3 ) @ B13 @ B6 ) ) ) ) ).
% bij_betw_inv_into_subset
thf(fact_6715_inv__unique__comp,axiom,
! [B: $tType,A: $tType,F3: B > A,G3: A > B] :
( ( ( comp @ B @ A @ A @ F3 @ G3 )
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ B @ G3 @ F3 )
= ( id @ B ) )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 )
= G3 ) ) ) ).
% inv_unique_comp
thf(fact_6716_o__inv__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B,G3: C > A] :
( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( bij_betw @ C @ A @ G3 @ ( top_top @ ( set @ C ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( hilbert_inv_into @ C @ B @ ( top_top @ ( set @ C ) ) @ ( comp @ A @ B @ C @ F3 @ G3 ) )
= ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ ( top_top @ ( set @ C ) ) @ G3 ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) ) ) ) ) ).
% o_inv_distrib
thf(fact_6717_mono__inv,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B] :
( ( order_mono @ A @ B @ F3 )
=> ( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( order_mono @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) ) ) ) ) ).
% mono_inv
thf(fact_6718_inv__fn,axiom,
! [A: $tType,F3: A > A,N: nat] :
( ( bij_betw @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( compow @ ( A > A ) @ N @ F3 ) )
= ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) ) ) ) ).
% inv_fn
thf(fact_6719_bij__image__Collect__eq,axiom,
! [A: $tType,B: $tType,F3: A > B,P2: A > $o] :
( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F3 @ ( collect @ A @ P2 ) )
= ( collect @ B
@ ^ [Y4: B] : ( P2 @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 @ Y4 ) ) ) ) ) ).
% bij_image_Collect_eq
thf(fact_6720_surj__iff,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ( comp @ B @ A @ A @ F3 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F3 ) )
= ( id @ A ) ) ) ).
% surj_iff
thf(fact_6721_inj__imp__bij__betw__inv,axiom,
! [B: $tType,A: $tType,F3: A > B,M5: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( image2 @ A @ B @ F3 @ M5 ) @ M5 ) ) ).
% inj_imp_bij_betw_inv
thf(fact_6722_inj__iff,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
= ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ F3 )
= ( id @ A ) ) ) ).
% inj_iff
thf(fact_6723_bij__vimage__eq__inv__image,axiom,
! [A: $tType,B: $tType,F3: A > B,A6: set @ B] :
( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( vimage @ A @ B @ F3 @ A6 )
= ( image2 @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ A6 ) ) ) ).
% bij_vimage_eq_inv_image
thf(fact_6724_fn__o__inv__fn__is__id,axiom,
! [A: $tType,F3: A > A,N: nat] :
( ( bij_betw @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) ) )
= ( ^ [X4: A] : X4 ) ) ) ).
% fn_o_inv_fn_is_id
thf(fact_6725_inv__fn__o__fn__is__id,axiom,
! [A: $tType,F3: A > A,N: nat] :
( ( bij_betw @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) ) @ ( compow @ ( A > A ) @ N @ F3 ) )
= ( ^ [X4: A] : X4 ) ) ) ).
% inv_fn_o_fn_is_id
thf(fact_6726_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_6727_iteratesp__def,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F4: A > A] :
( complete_lattice_lfp @ ( A > $o )
@ ^ [P5: A > $o,X4: A] :
( ? [Y4: A] :
( ( X4
= ( F4 @ Y4 ) )
& ( P5 @ Y4 ) )
| ? [M9: set @ A] :
( ( X4
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y4: A] :
( ( member @ A @ Y4 @ M9 )
=> ( P5 @ Y4 ) ) ) ) ) ) ) ) ).
% iteratesp_def
thf(fact_6728_iteratesp_OSup,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [M5: set @ A,F3: A > A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ M5 )
=> ( comple7512665784863727008ratesp @ A @ F3 @ X3 ) )
=> ( comple7512665784863727008ratesp @ A @ F3 @ ( complete_Sup_Sup @ A @ M5 ) ) ) ) ) ).
% iteratesp.Sup
thf(fact_6729_iteratesp_Ocases,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A,A3: A] :
( ( comple7512665784863727008ratesp @ A @ F3 @ A3 )
=> ( ! [X3: A] :
( ( A3
= ( F3 @ X3 ) )
=> ~ ( comple7512665784863727008ratesp @ A @ F3 @ X3 ) )
=> ~ ! [M8: set @ A] :
( ( A3
= ( complete_Sup_Sup @ A @ M8 ) )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ M8 )
=> ( comple7512665784863727008ratesp @ A @ F3 @ X5 ) ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_6730_iteratesp_Osimps,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F4: A > A,A7: A] :
( ? [X4: A] :
( ( A7
= ( F4 @ X4 ) )
& ( comple7512665784863727008ratesp @ A @ F4 @ X4 ) )
| ? [M9: set @ A] :
( ( A7
= ( 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_6731_strict__mono__inv__on__range,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( strict_mono_on @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% strict_mono_inv_on_range
thf(fact_6732_bijection_Oinv__comp__left,axiom,
! [A: $tType,F3: A > A] :
( ( hilbert_bijection @ A @ F3 )
=> ( ( comp @ A @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) @ F3 )
= ( id @ A ) ) ) ).
% bijection.inv_comp_left
thf(fact_6733_strict__mono__on__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F3: A > B,A6: set @ A,X: A,Y: A] :
( ( strict_mono_on @ A @ B @ F3 @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( member @ A @ Y @ A6 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_6734_bijection_OeqI,axiom,
! [A: $tType,F3: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F3 )
=> ( ( ( F3 @ A3 )
= ( F3 @ B2 ) )
=> ( A3 = B2 ) ) ) ).
% bijection.eqI
thf(fact_6735_bijection_Oeq__iff,axiom,
! [A: $tType,F3: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F3 )
=> ( ( ( F3 @ A3 )
= ( F3 @ B2 ) )
= ( A3 = B2 ) ) ) ).
% bijection.eq_iff
thf(fact_6736_bijection_Osurj,axiom,
! [A: $tType,F3: A > A] :
( ( hilbert_bijection @ A @ F3 )
=> ( ( image2 @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% bijection.surj
thf(fact_6737_bijection_Oinj,axiom,
! [A: $tType,F3: A > A] :
( ( hilbert_bijection @ A @ F3 )
=> ( inj_on @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.inj
thf(fact_6738_bijection__def,axiom,
! [A: $tType] :
( ( hilbert_bijection @ A )
= ( ^ [F4: A > A] : ( bij_betw @ A @ A @ F4 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% bijection_def
thf(fact_6739_bijection_Ointro,axiom,
! [A: $tType,F3: A > A] :
( ( bij_betw @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( hilbert_bijection @ A @ F3 ) ) ).
% bijection.intro
thf(fact_6740_bijection_Obij,axiom,
! [A: $tType,F3: A > A] :
( ( hilbert_bijection @ A @ F3 )
=> ( bij_betw @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.bij
thf(fact_6741_bijection_Oinv__right__eq__iff,axiom,
! [A: $tType,F3: A > A,B2: A,A3: A] :
( ( hilbert_bijection @ A @ F3 )
=> ( ( B2
= ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A3 ) )
= ( ( F3 @ B2 )
= A3 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_6742_bijection_Oinv__left__eq__iff,axiom,
! [A: $tType,F3: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F3 )
=> ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A3 )
= B2 )
= ( ( F3 @ B2 )
= A3 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_6743_bijection_Oeq__inv__iff,axiom,
! [A: $tType,F3: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F3 )
=> ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A3 )
= ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ B2 ) )
= ( A3 = B2 ) ) ) ).
% bijection.eq_inv_iff
thf(fact_6744_bijection_Oinv__right,axiom,
! [A: $tType,F3: A > A,A3: A] :
( ( hilbert_bijection @ A @ F3 )
=> ( ( F3 @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A3 ) )
= A3 ) ) ).
% bijection.inv_right
thf(fact_6745_bijection_Oinv__left,axiom,
! [A: $tType,F3: A > A,A3: A] :
( ( hilbert_bijection @ A @ F3 )
=> ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ ( F3 @ A3 ) )
= A3 ) ) ).
% bijection.inv_left
thf(fact_6746_bijection_Oeq__invI,axiom,
! [A: $tType,F3: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F3 )
=> ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ A3 )
= ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 @ B2 ) )
=> ( A3 = B2 ) ) ) ).
% bijection.eq_invI
thf(fact_6747_bijection_Osurj__inv,axiom,
! [A: $tType,F3: A > A] :
( ( hilbert_bijection @ A @ F3 )
=> ( ( image2 @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% bijection.surj_inv
thf(fact_6748_bijection_Oinj__inv,axiom,
! [A: $tType,F3: A > A] :
( ( hilbert_bijection @ A @ F3 )
=> ( inj_on @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.inj_inv
thf(fact_6749_bijection_Obij__inv,axiom,
! [A: $tType,F3: A > A] :
( ( hilbert_bijection @ A @ F3 )
=> ( bij_betw @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.bij_inv
thf(fact_6750_bijection_Oinv__comp__right,axiom,
! [A: $tType,F3: A > A] :
( ( hilbert_bijection @ A @ F3 )
=> ( ( comp @ A @ A @ A @ F3 @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F3 ) )
= ( id @ A ) ) ) ).
% bijection.inv_comp_right
thf(fact_6751_admissible__chfin,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P2: 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 ) @ P2 ) ) ) ).
% admissible_chfin
thf(fact_6752_VEBT__internal_Ooption__shift_Opelims,axiom,
! [A: $tType,X: A > A > A,Xa2: option @ A,Xb: option @ A,Y: option @ A] :
( ( ( vEBT_V2048590022279873568_shift @ A @ X @ Xa2 @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ Xa2 @ Xb ) ) )
=> ( ( ( Xa2
= ( none @ A ) )
=> ( ( Y
= ( none @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Xb ) ) ) ) )
=> ( ! [V2: A] :
( ( Xa2
= ( some @ A @ V2 ) )
=> ( ( Xb
= ( none @ A ) )
=> ( ( Y
= ( none @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V2 ) @ ( none @ A ) ) ) ) ) ) )
=> ~ ! [A5: A] :
( ( Xa2
= ( some @ A @ A5 ) )
=> ! [B4: A] :
( ( Xb
= ( some @ A @ B4 ) )
=> ( ( Y
= ( some @ A @ ( X @ A5 @ B4 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V459564278314245337ft_rel @ A ) @ ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A5 ) @ ( some @ A @ B4 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.pelims
thf(fact_6753_admissible__disj,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P2: A > $o,Q: A > $o] :
( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P2 )
=> ( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ Q )
=> ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A )
@ ^ [X4: A] :
( ( P2 @ X4 )
| ( Q @ X4 ) ) ) ) ) ) ).
% admissible_disj
thf(fact_6754_Domain__insert,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 ) )
= ( insert @ A @ A3 @ ( domain @ A @ B @ R3 ) ) ) ).
% Domain_insert
thf(fact_6755_ndepth__Push__Node__aux,axiom,
! [A: $tType,I2: nat,F3: nat > ( sum_sum @ A @ nat ),K2: nat] :
( ( ( case_nat @ ( sum_sum @ A @ nat ) @ ( sum_Inr @ nat @ A @ ( suc @ I2 ) ) @ F3 @ K2 )
= ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) )
=> ( ord_less_eq @ nat
@ ( suc
@ ( ord_Least @ nat
@ ^ [X4: nat] :
( ( F3 @ X4 )
= ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) ) ) )
@ K2 ) ) ).
% ndepth_Push_Node_aux
thf(fact_6756_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
=> ~ ! [B4: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B4 ) @ R3 ) ) ).
% Domain.cases
thf(fact_6757_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
= ( ? [A7: A,B5: B] :
( ( A3 = A7 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R3 ) ) ) ) ).
% Domain.simps
thf(fact_6758_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) ) ) ).
% Domain.DomainI
thf(fact_6759_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
=> ~ ! [B4: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B4 ) @ R3 ) ) ).
% DomainE
thf(fact_6760_Domain__iff,axiom,
! [A: $tType,B: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
= ( ? [Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y4 ) @ R3 ) ) ) ).
% Domain_iff
thf(fact_6761_Not__Domain__rtrancl,axiom,
! [A: $tType,X: A,R2: set @ ( product_prod @ A @ A ),Y: A] :
( ~ ( member @ A @ X @ ( domain @ A @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_6762_Domain__unfold,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ^ [X4: A] :
? [Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R ) ) ) ) ).
% Domain_unfold
thf(fact_6763_sum_Osize_I4_J,axiom,
! [B: $tType,A: $tType,X2: B] :
( ( size_size @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A @ X2 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% sum.size(4)
thf(fact_6764_sum_Osize__gen_I2_J,axiom,
! [A: $tType,B: $tType,Xa2: A > nat,X: B > nat,X2: B] :
( ( basic_BNF_size_sum @ A @ B @ Xa2 @ X @ ( sum_Inr @ B @ A @ X2 ) )
= ( plus_plus @ nat @ ( X @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% sum.size_gen(2)
thf(fact_6765_Node__def,axiom,
! [A: $tType,B: $tType] :
( ( old_Node @ B @ A )
= ( collect @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) )
@ ^ [P5: product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )] :
? [F4: nat > ( sum_sum @ B @ nat ),X4: sum_sum @ A @ nat,K3: nat] :
( ( P5
= ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ F4 @ X4 ) )
& ( ( F4 @ K3 )
= ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) ) ) ) ).
% Node_def
thf(fact_6766_Node__K0__I,axiom,
! [B: $tType,A: $tType,A3: sum_sum @ B @ nat] :
( member @ ( product_prod @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat ) )
@ ( product_Pair @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat )
@ ^ [K3: nat] : ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) )
@ A3 )
@ ( old_Node @ A @ B ) ) ).
% Node_K0_I
thf(fact_6767_Push__neq__K0,axiom,
! [A: $tType,K2: nat,F3: nat > ( sum_sum @ A @ nat )] :
( ( old_Push @ A @ ( sum_Inr @ nat @ A @ ( suc @ K2 ) ) @ F3 )
!= ( ^ [Z4: nat] : ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) ) ) ).
% Push_neq_K0
thf(fact_6768_ndepth__def,axiom,
! [B: $tType,A: $tType] :
( ( old_ndepth @ A @ B )
= ( ^ [N5: old_node @ A @ B] :
( product_case_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ nat
@ ^ [F4: nat > ( sum_sum @ B @ nat ),X4: sum_sum @ A @ nat] :
( ord_Least @ nat
@ ^ [K3: nat] :
( ( F4 @ K3 )
= ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) )
@ ( old_Rep_Node @ A @ B @ N5 ) ) ) ) ).
% ndepth_def
thf(fact_6769_ndepth__K0,axiom,
! [A: $tType,B: $tType,X: sum_sum @ A @ nat] :
( ( old_ndepth @ A @ B
@ ( old_Abs_Node @ B @ A
@ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
@ ^ [K3: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
@ X ) ) )
= ( zero_zero @ nat ) ) ).
% ndepth_K0
thf(fact_6770_ndepth__Push__Node,axiom,
! [B: $tType,A: $tType,I2: nat,N: old_node @ A @ B] :
( ( old_ndepth @ A @ B @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ I2 ) ) @ N ) )
= ( suc @ ( old_ndepth @ A @ B @ N ) ) ) ).
% ndepth_Push_Node
thf(fact_6771_Atom__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Atom @ A @ B )
= ( ^ [X4: sum_sum @ A @ nat] :
( insert @ ( old_node @ A @ B )
@ ( old_Abs_Node @ B @ A
@ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
@ ^ [K3: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
@ X4 ) )
@ ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) ).
% Atom_def
thf(fact_6772_Scons__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Scons @ A @ B )
= ( ^ [M9: set @ ( old_node @ A @ B ),N6: set @ ( old_node @ A @ B )] : ( sup_sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( one_one @ nat ) ) ) @ M9 ) @ ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ ( one_one @ nat ) ) ) ) @ N6 ) ) ) ) ).
% Scons_def
thf(fact_6773_ntrunc__0,axiom,
! [B: $tType,A: $tType,M5: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( zero_zero @ nat ) @ M5 )
= ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% ntrunc_0
thf(fact_6774_int__encode__def,axiom,
( nat_int_encode
= ( ^ [I: int] : ( nat_sum_encode @ ( if @ ( sum_sum @ nat @ nat ) @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I ) @ ( sum_Inl @ nat @ nat @ ( nat2 @ I ) ) @ ( sum_Inr @ nat @ nat @ ( nat2 @ ( minus_minus @ int @ ( uminus_uminus @ int @ I ) @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% int_encode_def
thf(fact_6775_ntrunc__Scons,axiom,
! [B: $tType,A: $tType,K2: nat,M5: set @ ( old_node @ A @ B ),N7: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ K2 ) @ ( old_Scons @ A @ B @ M5 @ N7 ) )
= ( old_Scons @ A @ B @ ( old_ntrunc @ A @ B @ K2 @ M5 ) @ ( old_ntrunc @ A @ B @ K2 @ N7 ) ) ) ).
% ntrunc_Scons
thf(fact_6776_ntrunc__Atom,axiom,
! [B: $tType,A: $tType,K2: nat,A3: sum_sum @ A @ nat] :
( ( old_ntrunc @ A @ B @ ( suc @ K2 ) @ ( old_Atom @ A @ B @ A3 ) )
= ( old_Atom @ A @ B @ A3 ) ) ).
% ntrunc_Atom
thf(fact_6777_surj__sum__encode,axiom,
( ( image2 @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_sum_encode
thf(fact_6778_inj__sum__encode,axiom,
! [A6: set @ ( sum_sum @ nat @ nat )] : ( inj_on @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ A6 ) ).
% inj_sum_encode
thf(fact_6779_sum__encode__eq,axiom,
! [X: sum_sum @ nat @ nat,Y: sum_sum @ nat @ nat] :
( ( ( nat_sum_encode @ X )
= ( nat_sum_encode @ Y ) )
= ( X = Y ) ) ).
% sum_encode_eq
thf(fact_6780_bij__sum__encode,axiom,
bij_betw @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_sum_encode
thf(fact_6781_le__sum__encode__Inl,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq @ nat @ X @ Y )
=> ( ord_less_eq @ nat @ X @ ( nat_sum_encode @ ( sum_Inl @ nat @ nat @ Y ) ) ) ) ).
% le_sum_encode_Inl
thf(fact_6782_le__sum__encode__Inr,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq @ nat @ X @ Y )
=> ( ord_less_eq @ nat @ X @ ( nat_sum_encode @ ( sum_Inr @ nat @ nat @ Y ) ) ) ) ).
% le_sum_encode_Inr
thf(fact_6783_sum_Osize_I3_J,axiom,
! [A: $tType,B: $tType,X1: A] :
( ( size_size @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ X1 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% sum.size(3)
thf(fact_6784_sum_Osize__gen_I1_J,axiom,
! [B: $tType,A: $tType,Xa2: A > nat,X: B > nat,X1: A] :
( ( basic_BNF_size_sum @ A @ B @ Xa2 @ X @ ( sum_Inl @ A @ B @ X1 ) )
= ( plus_plus @ nat @ ( Xa2 @ X1 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% sum.size_gen(1)
thf(fact_6785_sum__decode__def,axiom,
( nat_sum_decode
= ( ^ [N5: nat] : ( if @ ( sum_sum @ nat @ nat ) @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( sum_Inl @ nat @ nat @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( sum_Inr @ nat @ nat @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sum_decode_def
thf(fact_6786_ntrunc__one__In0,axiom,
! [B: $tType,A: $tType,M5: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ ( zero_zero @ nat ) ) @ ( old_In0 @ A @ B @ M5 ) )
= ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% ntrunc_one_In0
thf(fact_6787_sum__encode__inverse,axiom,
! [X: sum_sum @ nat @ nat] :
( ( nat_sum_decode @ ( nat_sum_encode @ X ) )
= X ) ).
% sum_encode_inverse
thf(fact_6788_sum__decode__inverse,axiom,
! [N: nat] :
( ( nat_sum_encode @ ( nat_sum_decode @ N ) )
= N ) ).
% sum_decode_inverse
thf(fact_6789_ntrunc__In0,axiom,
! [B: $tType,A: $tType,K2: nat,M5: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ ( suc @ K2 ) ) @ ( old_In0 @ A @ B @ M5 ) )
= ( old_In0 @ A @ B @ ( old_ntrunc @ A @ B @ ( suc @ K2 ) @ M5 ) ) ) ).
% ntrunc_In0
thf(fact_6790_bij__sum__decode,axiom,
bij_betw @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) ).
% bij_sum_decode
thf(fact_6791_sum__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( ( nat_sum_decode @ X )
= ( nat_sum_decode @ Y ) )
= ( X = Y ) ) ).
% sum_decode_eq
thf(fact_6792_inj__sum__decode,axiom,
! [A6: set @ nat] : ( inj_on @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ A6 ) ).
% inj_sum_decode
thf(fact_6793_surj__sum__decode,axiom,
( ( image2 @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) ) ).
% surj_sum_decode
thf(fact_6794_nth__item_Opinduct,axiom,
! [A0: nat,P2: nat > $o] :
( ( accp @ nat @ nth_item_rel @ A0 )
=> ( ( ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) )
=> ( P2 @ ( zero_zero @ nat ) ) )
=> ( ! [N3: nat] :
( ( accp @ nat @ nth_item_rel @ ( suc @ N3 ) )
=> ( ! [A9: nat,Aa3: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inl @ nat @ nat @ A9 ) )
=> ( ( ( nat_sum_decode @ A9 )
= ( sum_Inl @ nat @ nat @ Aa3 ) )
=> ( P2 @ Aa3 ) ) )
=> ( ! [A9: nat,B7: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inl @ nat @ nat @ A9 ) )
=> ( ( ( nat_sum_decode @ A9 )
= ( sum_Inr @ nat @ nat @ B7 ) )
=> ( P2 @ B7 ) ) )
=> ( ! [B7: nat,Ba2: nat,X5: nat,Y6: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inr @ nat @ nat @ B7 ) )
=> ( ( ( nat_sum_decode @ B7 )
= ( sum_Inr @ nat @ nat @ Ba2 ) )
=> ( ( ( product_Pair @ nat @ nat @ X5 @ Y6 )
= ( nat_prod_decode @ Ba2 ) )
=> ( P2 @ X5 ) ) ) )
=> ( ! [B7: nat,Ba2: nat,X5: nat,Y6: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inr @ nat @ nat @ B7 ) )
=> ( ( ( nat_sum_decode @ B7 )
= ( sum_Inr @ nat @ nat @ Ba2 ) )
=> ( ( ( product_Pair @ nat @ nat @ X5 @ Y6 )
= ( nat_prod_decode @ Ba2 ) )
=> ( P2 @ Y6 ) ) ) )
=> ( P2 @ ( suc @ N3 ) ) ) ) ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% nth_item.pinduct
thf(fact_6795_ntrunc__one__In1,axiom,
! [B: $tType,A: $tType,M5: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ ( zero_zero @ nat ) ) @ ( old_In1 @ A @ B @ M5 ) )
= ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% ntrunc_one_In1
thf(fact_6796_ntrunc__In1,axiom,
! [B: $tType,A: $tType,K2: nat,M5: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ ( suc @ K2 ) ) @ ( old_In1 @ A @ B @ M5 ) )
= ( old_In1 @ A @ B @ ( old_ntrunc @ A @ B @ ( suc @ K2 ) @ M5 ) ) ) ).
% ntrunc_In1
thf(fact_6797_int__decode__def,axiom,
( nat_int_decode
= ( ^ [N5: nat] :
( sum_case_sum @ nat @ int @ nat @ ( semiring_1_of_nat @ int )
@ ^ [B5: nat] : ( minus_minus @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ B5 ) ) @ ( one_one @ int ) )
@ ( nat_sum_decode @ N5 ) ) ) ) ).
% int_decode_def
thf(fact_6798_sum__encode__def,axiom,
( nat_sum_encode
= ( sum_case_sum @ nat @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ^ [B5: nat] : ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B5 ) ) ) ) ).
% sum_encode_def
thf(fact_6799_ntrunc__Leaf,axiom,
! [B: $tType,A: $tType,K2: nat,A3: A] :
( ( old_ntrunc @ A @ B @ ( suc @ K2 ) @ ( old_Leaf @ A @ B @ A3 ) )
= ( old_Leaf @ A @ B @ A3 ) ) ).
% ntrunc_Leaf
thf(fact_6800_In0__def,axiom,
! [A: $tType,B: $tType] :
( ( old_In0 @ A @ B )
= ( old_Scons @ A @ B @ ( old_Numb @ A @ B @ ( zero_zero @ nat ) ) ) ) ).
% In0_def
thf(fact_6801_ntrunc__Numb,axiom,
! [A: $tType,B: $tType,K2: nat,I2: nat] :
( ( old_ntrunc @ A @ B @ ( suc @ K2 ) @ ( old_Numb @ A @ B @ I2 ) )
= ( old_Numb @ A @ B @ I2 ) ) ).
% ntrunc_Numb
thf(fact_6802_prod_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,B6: set @ C,G3: ( sum_sum @ B @ C ) > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ C @ B6 )
=> ( ( groups7121269368397514597t_prod @ ( sum_sum @ B @ C ) @ A @ G3 @ ( sum_Plus @ B @ C @ A6 @ B6 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G3 @ ( sum_Inl @ B @ C ) ) @ A6 ) @ ( groups7121269368397514597t_prod @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G3 @ ( sum_Inr @ C @ B ) ) @ B6 ) ) ) ) ) ) ).
% prod.Plus
thf(fact_6803_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,X: A,Y: A,Zs: list @ A] :
( ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs ) ) ) ) @ X @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_6804_card__Plus__conv__if,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ( ( ( finite_finite2 @ A @ A6 )
& ( finite_finite2 @ B @ B6 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B6 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B6 ) ) ) )
& ( ~ ( ( finite_finite2 @ A @ A6 )
& ( finite_finite2 @ B @ B6 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B6 ) )
= ( zero_zero @ nat ) ) ) ) ).
% card_Plus_conv_if
thf(fact_6805_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa2: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa2 )
= Y )
=> ( ! [X3: A] :
( ( Xa2
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( Y != X3 ) )
=> ( ! [X3: A,Y3: A,Zs2: list @ A] :
( ( Xa2
= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) )
=> ( Y
!= ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) )
=> ~ ( ( Xa2
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_6806_enat__def,axiom,
( extended_enat2
= ( ^ [N5: nat] : ( extended_Abs_enat @ ( some @ nat @ N5 ) ) ) ) ).
% enat_def
thf(fact_6807_option_Othe__def,axiom,
! [A: $tType] :
( ( the2 @ A )
= ( case_option @ A @ A @ ( undefined @ A )
@ ^ [X24: A] : X24 ) ) ).
% option.the_def
thf(fact_6808_nth__item_Opsimps_I1_J,axiom,
! [A: $tType] :
( ( countable @ A )
=> ( ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) )
=> ( ( nth_item @ A @ ( zero_zero @ nat ) )
= ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) ) ) ).
% nth_item.psimps(1)
thf(fact_6809_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa2: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa2 ) )
=> ( ! [X3: A] :
( ( Xa2
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ( Y = X3 )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X3: A,Y3: A,Zs2: list @ A] :
( ( Xa2
= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) )
=> ( ( Y
= ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) ) )
=> ~ ( ( 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 ) @ X @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
thf(fact_6810_nth__item_Osimps_I1_J,axiom,
! [A: $tType] :
( ( countable @ A )
=> ( ( nth_item @ A @ ( zero_zero @ nat ) )
= ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) ) ).
% nth_item.simps(1)
thf(fact_6811_nth__item_Opelims,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [X: nat,Y: set @ ( old_node @ A @ product_unit )] :
( ( ( nth_item @ A @ X )
= Y )
=> ( ( accp @ nat @ nth_item_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) )
=> ~ ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( ( Y
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ^ [J: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ( nat_sum_decode @ I ) )
@ ^ [I: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J ) )
@ ^ [J: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A7: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A7 ) @ ( nth_item @ A @ B5 ) )
@ ( nat_prod_decode @ J ) )
@ ( nat_sum_decode @ I ) )
@ ( nat_sum_decode @ N3 ) ) )
=> ~ ( accp @ nat @ nth_item_rel @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% nth_item.pelims
thf(fact_6812_nth__item_Opsimps_I2_J,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [N: nat] :
( ( accp @ nat @ nth_item_rel @ ( suc @ N ) )
=> ( ( nth_item @ A @ ( suc @ N ) )
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ^ [J: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ( nat_sum_decode @ I ) )
@ ^ [I: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J ) )
@ ^ [J: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A7: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A7 ) @ ( nth_item @ A @ B5 ) )
@ ( nat_prod_decode @ J ) )
@ ( nat_sum_decode @ I ) )
@ ( nat_sum_decode @ N ) ) ) ) ) ).
% nth_item.psimps(2)
thf(fact_6813_nth__item_Osimps_I2_J,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [N: nat] :
( ( nth_item @ A @ ( suc @ N ) )
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ^ [J: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ( nat_sum_decode @ I ) )
@ ^ [I: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J ) )
@ ^ [J: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A7: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A7 ) @ ( nth_item @ A @ B5 ) )
@ ( nat_prod_decode @ J ) )
@ ( nat_sum_decode @ I ) )
@ ( nat_sum_decode @ N ) ) ) ) ).
% nth_item.simps(2)
thf(fact_6814_nth__item_Oelims,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [X: nat,Y: set @ ( old_node @ A @ product_unit )] :
( ( ( nth_item @ A @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( Y
!= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ^ [J: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ( nat_sum_decode @ I ) )
@ ^ [I: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J ) )
@ ^ [J: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A7: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A7 ) @ ( nth_item @ A @ B5 ) )
@ ( nat_prod_decode @ J ) )
@ ( nat_sum_decode @ I ) )
@ ( nat_sum_decode @ N3 ) ) ) ) ) ) ) ).
% nth_item.elims
thf(fact_6815_Bseq__monoseq__convergent_H__dec,axiom,
! [F3: nat > real,M5: nat] :
( ( bfun @ nat @ real
@ ^ [N5: nat] : ( F3 @ ( plus_plus @ nat @ N5 @ M5 ) )
@ ( at_top @ nat ) )
=> ( ! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ M )
=> ( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ real @ ( F3 @ N3 ) @ ( F3 @ M ) ) ) )
=> ( topolo6863149650580417670ergent @ real @ F3 ) ) ) ).
% Bseq_monoseq_convergent'_dec
thf(fact_6816_Bseq__monoseq__convergent_H__inc,axiom,
! [F3: nat > real,M5: nat] :
( ( bfun @ nat @ real
@ ^ [N5: nat] : ( F3 @ ( plus_plus @ nat @ N5 @ M5 ) )
@ ( at_top @ nat ) )
=> ( ! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M5 @ M )
=> ( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ real @ ( F3 @ M ) @ ( F3 @ N3 ) ) ) )
=> ( topolo6863149650580417670ergent @ real @ F3 ) ) ) ).
% Bseq_monoseq_convergent'_inc
thf(fact_6817_lim__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,X: A] :
( ( topolo6863149650580417670ergent @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ X )
=> ( ord_less_eq @ A @ ( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat ) @ F3 ) @ X ) ) ) ) ).
% lim_le
thf(fact_6818_convergent__Suc__iff,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A] :
( ( topolo6863149650580417670ergent @ A
@ ^ [N5: nat] : ( F3 @ ( suc @ N5 ) ) )
= ( topolo6863149650580417670ergent @ A @ F3 ) ) ) ).
% convergent_Suc_iff
thf(fact_6819_convergent__mult,axiom,
! [A: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [X8: nat > A,Y7: nat > A] :
( ( topolo6863149650580417670ergent @ A @ X8 )
=> ( ( topolo6863149650580417670ergent @ A @ Y7 )
=> ( topolo6863149650580417670ergent @ A
@ ^ [N5: nat] : ( times_times @ A @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ) ).
% convergent_mult
thf(fact_6820_convergent__mult__const__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( topolo6863149650580417670ergent @ A
@ ^ [N5: nat] : ( times_times @ A @ C3 @ ( F3 @ N5 ) ) )
= ( topolo6863149650580417670ergent @ A @ F3 ) ) ) ) ).
% convergent_mult_const_iff
thf(fact_6821_convergent__mult__const__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( topolo6863149650580417670ergent @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F3 @ N5 ) @ C3 ) )
= ( topolo6863149650580417670ergent @ A @ F3 ) ) ) ) ).
% convergent_mult_const_right_iff
thf(fact_6822_Bseq__mono__convergent,axiom,
! [X8: nat > real] :
( ( bfun @ nat @ real @ X8 @ ( at_top @ nat ) )
=> ( ! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ real @ ( X8 @ M ) @ ( X8 @ N3 ) ) )
=> ( topolo6863149650580417670ergent @ real @ X8 ) ) ) ).
% Bseq_mono_convergent
thf(fact_6823_iterates_Osimps,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A3: A,F3: A > A] :
( ( member @ A @ A3 @ ( comple6359979572994053840erates @ A @ F3 ) )
= ( ? [X4: A] :
( ( A3
= ( F3 @ X4 ) )
& ( member @ A @ X4 @ ( comple6359979572994053840erates @ A @ F3 ) ) )
| ? [M9: set @ A] :
( ( A3
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [X4: A] :
( ( member @ A @ X4 @ M9 )
=> ( member @ A @ X4 @ ( comple6359979572994053840erates @ A @ F3 ) ) ) ) ) ) ) ).
% iterates.simps
thf(fact_6824_iterates_OSup,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [M5: set @ A,F3: A > A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ M5 )
=> ( member @ A @ X3 @ ( comple6359979572994053840erates @ A @ F3 ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ M5 ) @ ( comple6359979572994053840erates @ A @ F3 ) ) ) ) ) ).
% iterates.Sup
thf(fact_6825_iterates_Ocases,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A3: A,F3: A > A] :
( ( member @ A @ A3 @ ( comple6359979572994053840erates @ A @ F3 ) )
=> ( ! [X3: A] :
( ( A3
= ( F3 @ X3 ) )
=> ~ ( member @ A @ X3 @ ( comple6359979572994053840erates @ A @ F3 ) ) )
=> ~ ! [M8: set @ A] :
( ( A3
= ( complete_Sup_Sup @ A @ M8 ) )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ M8 )
=> ( member @ A @ X5 @ ( comple6359979572994053840erates @ A @ F3 ) ) ) ) ) ) ) ) ).
% iterates.cases
thf(fact_6826_chain__iterates,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F3 )
=> ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( comple6359979572994053840erates @ A @ F3 ) ) ) ) ).
% chain_iterates
thf(fact_6827_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_6828_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_6829_iterates__le__f,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A,F3: A > A] :
( ( member @ A @ X @ ( comple6359979572994053840erates @ A @ F3 ) )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F3 )
=> ( ord_less_eq @ A @ X @ ( F3 @ X ) ) ) ) ) ).
% iterates_le_f
thf(fact_6830_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_6831_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_6832_fixp__induct,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P2: A > $o,F3: A > A] :
( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P2 )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F3 )
=> ( ( P2 @ ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( P2 @ ( F3 @ X3 ) ) )
=> ( P2 @ ( comple115746919287870866o_fixp @ A @ F3 ) ) ) ) ) ) ) ).
% fixp_induct
thf(fact_6833_iterates__fixp,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F3 )
=> ( member @ A @ ( comple115746919287870866o_fixp @ A @ F3 ) @ ( comple6359979572994053840erates @ A @ F3 ) ) ) ) ).
% iterates_fixp
thf(fact_6834_fixp__lowerbound,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A,Z3: A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ Z3 ) @ Z3 )
=> ( ord_less_eq @ A @ ( comple115746919287870866o_fixp @ A @ F3 ) @ Z3 ) ) ) ) ).
% fixp_lowerbound
thf(fact_6835_fixp__unfold,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F3 )
=> ( ( comple115746919287870866o_fixp @ A @ F3 )
= ( F3 @ ( comple115746919287870866o_fixp @ A @ F3 ) ) ) ) ) ).
% fixp_unfold
thf(fact_6836_length__splice,axiom,
! [A: $tType,Xs2: list @ A,Ys: list @ A] :
( ( size_size @ ( list @ A ) @ ( splice @ A @ Xs2 @ Ys ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% length_splice
thf(fact_6837_map__conv__bind__option,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F4: B > A,X4: option @ B] : ( bind @ B @ A @ X4 @ ( comp @ A @ ( option @ A ) @ B @ ( some @ A ) @ F4 ) ) ) ) ).
% map_conv_bind_option
thf(fact_6838_bind__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,X: option @ C,F3: C > ( option @ B ),G3: B > ( option @ A )] :
( ( bind @ B @ A @ ( bind @ C @ B @ X @ F3 ) @ G3 )
= ( bind @ C @ A @ X
@ ^ [Y4: C] : ( bind @ B @ A @ ( F3 @ Y4 ) @ G3 ) ) ) ).
% bind_assoc
thf(fact_6839_bind__runit,axiom,
! [A: $tType,X: option @ A] :
( ( bind @ A @ A @ X @ ( some @ A ) )
= X ) ).
% bind_runit
thf(fact_6840_bind__rzero,axiom,
! [B: $tType,A: $tType,X: option @ B] :
( ( bind @ B @ A @ X
@ ^ [X4: B] : ( none @ A ) )
= ( none @ A ) ) ).
% bind_rzero
thf(fact_6841_bind__eq__None__conv,axiom,
! [B: $tType,A: $tType,A3: option @ B,F3: B > ( option @ A )] :
( ( ( bind @ B @ A @ A3 @ F3 )
= ( none @ A ) )
= ( ( A3
= ( none @ B ) )
| ( ( F3 @ ( the2 @ B @ A3 ) )
= ( none @ A ) ) ) ) ).
% bind_eq_None_conv
thf(fact_6842_bind_Obind__lzero,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B )] :
( ( bind @ A @ B @ ( none @ A ) @ F3 )
= ( none @ B ) ) ).
% bind.bind_lzero
thf(fact_6843_bind_Obind__lunit,axiom,
! [B: $tType,A: $tType,X: A,F3: A > ( option @ B )] :
( ( bind @ A @ B @ ( some @ A @ X ) @ F3 )
= ( F3 @ X ) ) ).
% bind.bind_lunit
thf(fact_6844_Option_Obind__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F3: A > ( option @ B ),G3: A > ( option @ B )] :
( ( X = Y )
=> ( ! [A5: A] :
( ( Y
= ( some @ A @ A5 ) )
=> ( ( F3 @ A5 )
= ( G3 @ A5 ) ) )
=> ( ( bind @ A @ B @ X @ F3 )
= ( bind @ A @ B @ Y @ G3 ) ) ) ) ).
% Option.bind_cong
thf(fact_6845_bind__eq__Some__conv,axiom,
! [A: $tType,B: $tType,F3: option @ B,G3: B > ( option @ A ),X: A] :
( ( ( bind @ B @ A @ F3 @ G3 )
= ( some @ A @ X ) )
= ( ? [Y4: B] :
( ( F3
= ( some @ B @ Y4 ) )
& ( ( G3 @ Y4 )
= ( some @ A @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_6846_bind__option__cong__code,axiom,
! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F3: A > ( option @ B )] :
( ( X = Y )
=> ( ( bind @ A @ B @ X @ F3 )
= ( bind @ A @ B @ Y @ F3 ) ) ) ).
% bind_option_cong_code
thf(fact_6847_bind__split,axiom,
! [A: $tType,B: $tType,P2: ( option @ A ) > $o,M2: option @ B,F3: B > ( option @ A )] :
( ( P2 @ ( bind @ B @ A @ M2 @ F3 ) )
= ( ( ( M2
= ( none @ B ) )
=> ( P2 @ ( none @ A ) ) )
& ! [V4: B] :
( ( M2
= ( some @ B @ V4 ) )
=> ( P2 @ ( F3 @ V4 ) ) ) ) ) ).
% bind_split
thf(fact_6848_bind__split__asm,axiom,
! [A: $tType,B: $tType,P2: ( option @ A ) > $o,M2: option @ B,F3: B > ( option @ A )] :
( ( P2 @ ( bind @ B @ A @ M2 @ F3 ) )
= ( ~ ( ( ( M2
= ( none @ B ) )
& ~ ( P2 @ ( none @ A ) ) )
| ? [X4: B] :
( ( M2
= ( some @ B @ X4 ) )
& ~ ( P2 @ ( F3 @ X4 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_6849_bind__map__option,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B,X: option @ C,G3: B > ( option @ A )] :
( ( bind @ B @ A @ ( map_option @ C @ B @ F3 @ X ) @ G3 )
= ( bind @ C @ A @ X @ ( comp @ B @ ( option @ A ) @ C @ G3 @ F3 ) ) ) ).
% bind_map_option
thf(fact_6850_map__option__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,X: option @ C,G3: C > ( option @ B )] :
( ( map_option @ B @ A @ F3 @ ( bind @ C @ B @ X @ G3 ) )
= ( bind @ C @ A @ X @ ( comp @ ( option @ B ) @ ( option @ A ) @ C @ ( map_option @ B @ A @ F3 ) @ G3 ) ) ) ).
% map_option_bind
thf(fact_6851_set__bind__option,axiom,
! [A: $tType,B: $tType,X: option @ B,F3: B > ( option @ A )] :
( ( set_option @ A @ ( bind @ B @ A @ X @ F3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( comp @ ( option @ A ) @ ( set @ A ) @ B @ ( set_option @ A ) @ F3 ) @ ( set_option @ B @ X ) ) ) ) ).
% set_bind_option
thf(fact_6852_continuous__attains__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S2: set @ A,F3: A > B] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ S2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S2 )
=> ( ord_less_eq @ B @ ( F3 @ Xa ) @ ( F3 @ X3 ) ) ) ) ) ) ) ) ).
% continuous_attains_sup
thf(fact_6853_elem__set,axiom,
! [A: $tType,X: A,Xo: option @ A] :
( ( member @ A @ X @ ( set_option @ A @ Xo ) )
= ( Xo
= ( some @ A @ X ) ) ) ).
% elem_set
thf(fact_6854_set__empty__eq,axiom,
! [A: $tType,Xo: option @ A] :
( ( ( set_option @ A @ Xo )
= ( bot_bot @ ( set @ A ) ) )
= ( Xo
= ( none @ A ) ) ) ).
% set_empty_eq
thf(fact_6855_bind__option__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F3: A > ( option @ B ),G3: A > ( option @ B )] :
( ( X = Y )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( set_option @ A @ Y ) )
=> ( ( F3 @ Z )
= ( G3 @ Z ) ) )
=> ( ( bind @ A @ B @ X @ F3 )
= ( bind @ A @ B @ Y @ G3 ) ) ) ) ).
% bind_option_cong
thf(fact_6856_option_Osimps_I14_J,axiom,
! [A: $tType] :
( ( set_option @ A @ ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% option.simps(14)
thf(fact_6857_ospec,axiom,
! [A: $tType,A6: option @ A,P2: A > $o,X: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_option @ A @ A6 ) )
=> ( P2 @ X3 ) )
=> ( ( A6
= ( some @ A @ X ) )
=> ( P2 @ X ) ) ) ).
% ospec
thf(fact_6858_option_Oset__intros,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( set_option @ A @ ( some @ A @ X2 ) ) ) ).
% option.set_intros
thf(fact_6859_option_Oset__cases,axiom,
! [A: $tType,E3: A,A3: option @ A] :
( ( member @ A @ E3 @ ( set_option @ A @ A3 ) )
=> ( A3
= ( some @ A @ E3 ) ) ) ).
% option.set_cases
thf(fact_6860_option_Omap__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Ya: option @ A,F3: A > B,G3: A > B] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( set_option @ A @ Ya ) )
=> ( ( F3 @ Z )
= ( G3 @ Z ) ) )
=> ( ( map_option @ A @ B @ F3 @ X )
= ( map_option @ A @ B @ G3 @ Ya ) ) ) ) ).
% option.map_cong
thf(fact_6861_option_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: option @ A,F3: A > B,G3: A > B] :
( ! [Z: A] :
( ( member @ A @ Z @ ( set_option @ A @ X ) )
=> ( ( F3 @ Z )
= ( G3 @ Z ) ) )
=> ( ( map_option @ A @ B @ F3 @ X )
= ( map_option @ A @ B @ G3 @ X ) ) ) ).
% option.map_cong0
thf(fact_6862_option_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: option @ A,Xa2: option @ A,F3: A > B,Fa: A > B] :
( ! [Z: A,Za2: A] :
( ( member @ A @ Z @ ( set_option @ A @ X ) )
=> ( ( member @ A @ Za2 @ ( set_option @ A @ Xa2 ) )
=> ( ( ( F3 @ Z )
= ( Fa @ Za2 ) )
=> ( Z = Za2 ) ) ) )
=> ( ( ( map_option @ A @ B @ F3 @ X )
= ( map_option @ A @ B @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% option.inj_map_strong
thf(fact_6863_map__option__idI,axiom,
! [A: $tType,X: option @ A,F3: A > A] :
( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set_option @ A @ X ) )
=> ( ( F3 @ Y3 )
= Y3 ) )
=> ( ( map_option @ A @ A @ F3 @ X )
= X ) ) ).
% map_option_idI
thf(fact_6864_option_Oset__sel,axiom,
! [A: $tType,A3: option @ A] :
( ( A3
!= ( none @ A ) )
=> ( member @ A @ ( the2 @ A @ A3 ) @ ( set_option @ A @ A3 ) ) ) ).
% option.set_sel
thf(fact_6865_option_Oset__map,axiom,
! [B: $tType,A: $tType,F3: A > B,V3: option @ A] :
( ( set_option @ B @ ( map_option @ A @ B @ F3 @ V3 ) )
= ( image2 @ A @ B @ F3 @ ( set_option @ A @ V3 ) ) ) ).
% option.set_map
thf(fact_6866_compact__attains__sup,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S3 )
=> ( ord_less_eq @ A @ Xa @ X3 ) ) ) ) ) ) ).
% compact_attains_sup
thf(fact_6867_compact__attains__inf,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S3 )
=> ( ord_less_eq @ A @ X3 @ Xa ) ) ) ) ) ) ).
% compact_attains_inf
thf(fact_6868_option_Osimps_I15_J,axiom,
! [A: $tType,X2: A] :
( ( set_option @ A @ ( some @ A @ X2 ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% option.simps(15)
thf(fact_6869_continuous__attains__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S2: set @ A,F3: A > B] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S2 @ F3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ S2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S2 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Xa ) ) ) ) ) ) ) ) ).
% continuous_attains_inf
thf(fact_6870_option_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,A7: option @ A,B5: option @ B] :
? [Z4: option @ ( product_prod @ A @ B )] :
( ( member @ ( option @ ( product_prod @ A @ B ) ) @ Z4
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X4: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z4 )
= A7 )
& ( ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z4 )
= B5 ) ) ) ) ).
% option.in_rel
thf(fact_6871_combine__options__def,axiom,
! [A: $tType] :
( ( combine_options @ A )
= ( ^ [F4: A > A > A,X4: option @ A,Y4: option @ A] :
( case_option @ ( option @ A ) @ A @ Y4
@ ^ [Z4: A] :
( case_option @ ( option @ A ) @ A @ ( some @ A @ Z4 )
@ ^ [Aa4: A] : ( some @ A @ ( F4 @ Z4 @ Aa4 ) )
@ Y4 )
@ X4 ) ) ) ).
% combine_options_def
thf(fact_6872_rel__option__None1,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,X: option @ B] :
( ( rel_option @ A @ B @ P2 @ ( none @ A ) @ X )
= ( X
= ( none @ B ) ) ) ).
% rel_option_None1
thf(fact_6873_rel__option__None2,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,X: option @ A] :
( ( rel_option @ A @ B @ P2 @ X @ ( none @ B ) )
= ( X
= ( none @ A ) ) ) ).
% rel_option_None2
thf(fact_6874_combine__options__simps_I3_J,axiom,
! [A: $tType,F3: A > A > A,A3: A,B2: A] :
( ( combine_options @ A @ F3 @ ( some @ A @ A3 ) @ ( some @ A @ B2 ) )
= ( some @ A @ ( F3 @ A3 @ B2 ) ) ) ).
% combine_options_simps(3)
thf(fact_6875_combine__options__simps_I1_J,axiom,
! [A: $tType,F3: A > A > A,Y: option @ A] :
( ( combine_options @ A @ F3 @ ( none @ A ) @ Y )
= Y ) ).
% combine_options_simps(1)
thf(fact_6876_combine__options__simps_I2_J,axiom,
! [A: $tType,F3: A > A > A,X: option @ A] :
( ( combine_options @ A @ F3 @ X @ ( none @ A ) )
= X ) ).
% combine_options_simps(2)
thf(fact_6877_rel__option__reflI,axiom,
! [A: $tType,Y: option @ A,P2: A > A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_option @ A @ Y ) )
=> ( P2 @ X3 @ X3 ) )
=> ( rel_option @ A @ A @ P2 @ Y @ Y ) ) ).
% rel_option_reflI
thf(fact_6878_option_Orel__refl__strong,axiom,
! [A: $tType,X: option @ A,Ra2: A > A > $o] :
( ! [Z: A] :
( ( member @ A @ Z @ ( set_option @ A @ X ) )
=> ( Ra2 @ Z @ Z ) )
=> ( rel_option @ A @ A @ Ra2 @ X @ X ) ) ).
% option.rel_refl_strong
thf(fact_6879_option_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X: option @ A,Y: option @ B,Ra2: A > B > $o] :
( ( rel_option @ A @ B @ R2 @ X @ Y )
=> ( ! [Z: A,Yb: B] :
( ( member @ A @ Z @ ( set_option @ A @ X ) )
=> ( ( member @ B @ Yb @ ( set_option @ B @ Y ) )
=> ( ( R2 @ Z @ Yb )
=> ( Ra2 @ Z @ Yb ) ) ) )
=> ( rel_option @ A @ B @ Ra2 @ X @ Y ) ) ) ).
% option.rel_mono_strong
thf(fact_6880_option_Orel__cong,axiom,
! [A: $tType,B: $tType,X: option @ A,Ya: option @ A,Y: option @ B,Xa2: option @ B,R2: A > B > $o,Ra2: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z: A,Yb: B] :
( ( member @ A @ Z @ ( set_option @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( set_option @ B @ Xa2 ) )
=> ( ( R2 @ Z @ Yb )
= ( Ra2 @ Z @ Yb ) ) ) )
=> ( ( rel_option @ A @ B @ R2 @ X @ Y )
= ( rel_option @ A @ B @ Ra2 @ Ya @ Xa2 ) ) ) ) ) ).
% option.rel_cong
thf(fact_6881_option_Orel__mono,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,Ra2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ Ra2 )
=> ( ord_less_eq @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ R2 ) @ ( rel_option @ A @ B @ Ra2 ) ) ) ).
% option.rel_mono
thf(fact_6882_option_Orec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S3: C > D > $o,R2: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S3 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R2 @ S3 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R2 ) @ S3 ) ) @ ( rec_option @ C @ A ) @ ( rec_option @ D @ B ) ) ).
% option.rec_transfer
thf(fact_6883_option_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I2: A > C,X: option @ A,Y: option @ B] :
( ( rel_option @ C @ B @ Sb @ ( map_option @ A @ C @ I2 @ X ) @ Y )
= ( rel_option @ A @ B
@ ^ [X4: A] : ( Sb @ ( I2 @ X4 ) )
@ X
@ Y ) ) ).
% option.rel_map(1)
thf(fact_6884_option_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: option @ A,G3: B > C,Y: option @ B] :
( ( rel_option @ A @ C @ Sa @ X @ ( map_option @ B @ C @ G3 @ Y ) )
= ( rel_option @ A @ B
@ ^ [X4: A,Y4: B] : ( Sa @ X4 @ ( G3 @ Y4 ) )
@ X
@ Y ) ) ).
% option.rel_map(2)
thf(fact_6885_option_Omap__transfer,axiom,
! [A: $tType,B: $tType,F: $tType,E: $tType,Rb2: A > E > $o,Sd: B > F > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F ) @ ( ( option @ A ) > ( option @ B ) ) @ ( ( option @ E ) > ( option @ F ) ) @ ( bNF_rel_fun @ A @ E @ B @ F @ Rb2 @ Sd ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ E ) @ ( option @ B ) @ ( option @ F ) @ ( rel_option @ A @ E @ Rb2 ) @ ( rel_option @ B @ F @ Sd ) ) @ ( map_option @ A @ B ) @ ( map_option @ E @ F ) ) ).
% option.map_transfer
thf(fact_6886_option_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] :
( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R2 )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ ^ [Option3: option @ A] :
( Option3
= ( none @ A ) )
@ ^ [Option3: option @ B] :
( Option3
= ( none @ B ) ) ) ).
% option.disc_transfer(1)
thf(fact_6887_option_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] :
( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R2 )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ ^ [Option3: option @ A] :
( Option3
!= ( none @ A ) )
@ ^ [Option3: option @ B] :
( Option3
!= ( none @ B ) ) ) ).
% option.disc_transfer(2)
thf(fact_6888_option_Octr__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( option @ A ) @ ( option @ B ) @ R2 @ ( rel_option @ A @ B @ R2 ) @ ( some @ A ) @ ( some @ B ) ) ).
% option.ctr_transfer(2)
thf(fact_6889_option_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( ( option @ C ) > ( option @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 ) )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ C ) @ ( ( option @ B ) > $o ) @ ( ( option @ D ) > $o ) @ ( rel_option @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( option @ B ) @ ( option @ D ) @ $o @ $o @ ( rel_option @ B @ D @ Sc )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 ) )
@ ( rel_option @ A @ B )
@ ( rel_option @ C @ D ) ) ).
% option.rel_transfer
thf(fact_6890_option_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ( bi_total @ A @ B @ R2 )
=> ( bi_total @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R2 ) ) ) ).
% option.bi_total_rel
thf(fact_6891_option_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X2: A,Y2: B] :
( ( rel_option @ A @ B @ R2 @ ( some @ A @ X2 ) @ ( some @ B @ Y2 ) )
= ( R2 @ X2 @ Y2 ) ) ).
% option.rel_inject(2)
thf(fact_6892_option_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X2: A,Y2: B] :
( ( R2 @ X2 @ Y2 )
=> ( rel_option @ A @ B @ R2 @ ( some @ A @ X2 ) @ ( some @ B @ Y2 ) ) ) ).
% option.rel_intros(2)
thf(fact_6893_option__rel__Some1,axiom,
! [A: $tType,B: $tType,A6: A > B > $o,X: A,Y: option @ B] :
( ( rel_option @ A @ B @ A6 @ ( some @ A @ X ) @ Y )
= ( ? [Y8: B] :
( ( Y
= ( some @ B @ Y8 ) )
& ( A6 @ X @ Y8 ) ) ) ) ).
% option_rel_Some1
thf(fact_6894_option__rel__Some2,axiom,
! [B: $tType,A: $tType,A6: A > B > $o,X: option @ A,Y: B] :
( ( rel_option @ A @ B @ A6 @ X @ ( some @ B @ Y ) )
= ( ? [X9: A] :
( ( X
= ( some @ A @ X9 ) )
& ( A6 @ X9 @ Y ) ) ) ) ).
% option_rel_Some2
thf(fact_6895_option_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] : ( rel_option @ A @ B @ R2 @ ( none @ A ) @ ( none @ B ) ) ).
% option.ctr_transfer(1)
thf(fact_6896_option_Orel__induct,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X: option @ A,Y: option @ B,Q: ( option @ A ) > ( option @ B ) > $o] :
( ( rel_option @ A @ B @ R2 @ X @ Y )
=> ( ( Q @ ( none @ A ) @ ( none @ B ) )
=> ( ! [A22: A,B24: B] :
( ( R2 @ A22 @ B24 )
=> ( Q @ ( some @ A @ A22 ) @ ( some @ B @ B24 ) ) )
=> ( Q @ X @ Y ) ) ) ) ).
% option.rel_induct
thf(fact_6897_option_Orel__cases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A3: option @ A,B2: option @ B] :
( ( rel_option @ A @ B @ R2 @ A3 @ B2 )
=> ( ( ( A3
= ( none @ A ) )
=> ( B2
!= ( none @ B ) ) )
=> ~ ! [X3: A] :
( ( A3
= ( some @ A @ X3 ) )
=> ! [Y3: B] :
( ( B2
= ( some @ B @ Y3 ) )
=> ~ ( R2 @ X3 @ Y3 ) ) ) ) ) ).
% option.rel_cases
thf(fact_6898_option_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,Y2: B] :
~ ( rel_option @ A @ B @ R2 @ ( none @ A ) @ ( some @ B @ Y2 ) ) ).
% option.rel_distinct(1)
thf(fact_6899_option_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,Y2: A] :
~ ( rel_option @ A @ B @ R2 @ ( some @ A @ Y2 ) @ ( none @ B ) ) ).
% option.rel_distinct(2)
thf(fact_6900_option_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,A7: option @ A,B5: option @ B] :
( ( ( A7
= ( none @ A ) )
= ( B5
= ( none @ B ) ) )
& ( ( A7
!= ( none @ A ) )
=> ( ( B5
!= ( none @ B ) )
=> ( R6 @ ( the2 @ A @ A7 ) @ ( the2 @ B @ B5 ) ) ) ) ) ) ) ).
% option.rel_sel
thf(fact_6901_option_Orel__eq,axiom,
! [A: $tType] :
( ( rel_option @ A @ A
@ ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [Y5: option @ A,Z2: option @ A] : Y5 = Z2 ) ) ).
% option.rel_eq
thf(fact_6902_option_Orel__refl,axiom,
! [B: $tType,Ra2: B > B > $o,X: option @ B] :
( ! [X3: B] : ( Ra2 @ X3 @ X3 )
=> ( rel_option @ B @ B @ Ra2 @ X @ X ) ) ).
% option.rel_refl
thf(fact_6903_combine__options__assoc,axiom,
! [A: $tType,F3: A > A > A,X: option @ A,Y: option @ A,Z3: option @ A] :
( ! [X3: A,Y3: A,Z: A] :
( ( F3 @ ( F3 @ X3 @ Y3 ) @ Z )
= ( F3 @ X3 @ ( F3 @ Y3 @ Z ) ) )
=> ( ( combine_options @ A @ F3 @ ( combine_options @ A @ F3 @ X @ Y ) @ Z3 )
= ( combine_options @ A @ F3 @ X @ ( combine_options @ A @ F3 @ Y @ Z3 ) ) ) ) ).
% combine_options_assoc
thf(fact_6904_combine__options__commute,axiom,
! [A: $tType,F3: A > A > A,X: option @ A,Y: option @ A] :
( ! [X3: A,Y3: A] :
( ( F3 @ X3 @ Y3 )
= ( F3 @ Y3 @ X3 ) )
=> ( ( combine_options @ A @ F3 @ X @ Y )
= ( combine_options @ A @ F3 @ Y @ X ) ) ) ).
% combine_options_commute
thf(fact_6905_combine__options__left__commute,axiom,
! [A: $tType,F3: A > A > A,Y: option @ A,X: option @ A,Z3: option @ A] :
( ! [X3: A,Y3: A] :
( ( F3 @ X3 @ Y3 )
= ( F3 @ Y3 @ X3 ) )
=> ( ! [X3: A,Y3: A,Z: A] :
( ( F3 @ ( F3 @ X3 @ Y3 ) @ Z )
= ( F3 @ X3 @ ( F3 @ Y3 @ Z ) ) )
=> ( ( combine_options @ A @ F3 @ Y @ ( combine_options @ A @ F3 @ X @ Z3 ) )
= ( combine_options @ A @ F3 @ X @ ( combine_options @ A @ F3 @ Y @ Z3 ) ) ) ) ) ).
% combine_options_left_commute
thf(fact_6906_rel__option__inf,axiom,
! [B: $tType,A: $tType,A6: A > B > $o,B6: A > B > $o] :
( ( inf_inf @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ A6 ) @ ( rel_option @ A @ B @ B6 ) )
= ( rel_option @ A @ B @ ( inf_inf @ ( A > B > $o ) @ A6 @ B6 ) ) ) ).
% rel_option_inf
thf(fact_6907_option_Orel__transp,axiom,
! [A: $tType,R2: A > A > $o] :
( ( transp @ A @ R2 )
=> ( transp @ ( option @ A ) @ ( rel_option @ A @ A @ R2 ) ) ) ).
% option.rel_transp
thf(fact_6908_option_Orel__conversep,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ( rel_option @ B @ A @ ( conversep @ A @ B @ R2 ) )
= ( conversep @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R2 ) ) ) ).
% option.rel_conversep
thf(fact_6909_option_Orel__flip,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A3: option @ B,B2: option @ A] :
( ( rel_option @ B @ A @ ( conversep @ A @ B @ R2 ) @ A3 @ B2 )
= ( rel_option @ A @ B @ R2 @ B2 @ A3 ) ) ).
% option.rel_flip
thf(fact_6910_option_Ocase__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S3: C > D > $o,R2: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S3 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R2 @ S3 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R2 ) @ S3 ) ) @ ( case_option @ C @ A ) @ ( case_option @ D @ B ) ) ).
% option.case_transfer
thf(fact_6911_option__bind__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A6: A > B > $o,B6: C > D > $o] : ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ ( ( A > ( option @ C ) ) > ( option @ C ) ) @ ( ( B > ( option @ D ) ) > ( option @ D ) ) @ ( rel_option @ A @ B @ A6 ) @ ( bNF_rel_fun @ ( A > ( option @ C ) ) @ ( B > ( option @ D ) ) @ ( option @ C ) @ ( option @ D ) @ ( bNF_rel_fun @ A @ B @ ( option @ C ) @ ( option @ D ) @ A6 @ ( rel_option @ C @ D @ B6 ) ) @ ( rel_option @ C @ D @ B6 ) ) @ ( bind @ A @ C ) @ ( bind @ B @ D ) ) ).
% option_bind_transfer
thf(fact_6912_rel__option__iff,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,X4: option @ A,Y4: option @ B] :
( product_case_prod @ ( option @ A ) @ ( option @ B ) @ $o
@ ^ [A7: option @ A,B5: option @ B] :
( case_option @ $o @ A
@ ( case_option @ $o @ B @ $true
@ ^ [C4: B] : $false
@ B5 )
@ ^ [Z4: A] : ( case_option @ $o @ B @ $false @ ( R6 @ Z4 ) @ B5 )
@ A7 )
@ ( product_Pair @ ( option @ A ) @ ( option @ B ) @ X4 @ Y4 ) ) ) ) ).
% rel_option_iff
thf(fact_6913_option_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o] :
( relcompp @ ( option @ A ) @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
@ ( conversep @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
@ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X4: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X4: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% option.rel_compp_Grp
thf(fact_6914_cut__def,axiom,
! [B: $tType,A: $tType] :
( ( cut @ A @ B )
= ( ^ [F4: A > B,R6: set @ ( product_prod @ A @ A ),X4: A,Y4: A] : ( if @ B @ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R6 ) @ ( F4 @ Y4 ) @ ( undefined @ B ) ) ) ) ).
% cut_def
thf(fact_6915_option_Orel__Grp,axiom,
! [B: $tType,A: $tType,A6: set @ A,F3: A > B] :
( ( rel_option @ A @ B @ ( bNF_Grp @ A @ B @ A6 @ F3 ) )
= ( bNF_Grp @ ( option @ A ) @ ( option @ B )
@ ( collect @ ( option @ A )
@ ^ [X4: option @ A] : ( ord_less_eq @ ( set @ A ) @ ( set_option @ A @ X4 ) @ A6 ) )
@ ( map_option @ A @ B @ F3 ) ) ) ).
% option.rel_Grp
thf(fact_6916_option_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: A > B > $o,S3: B > C > $o] :
( ( rel_option @ A @ C @ ( relcompp @ A @ B @ C @ R2 @ S3 ) )
= ( relcompp @ ( option @ A ) @ ( option @ B ) @ ( option @ C ) @ ( rel_option @ A @ B @ R2 ) @ ( rel_option @ B @ C @ S3 ) ) ) ).
% option.rel_compp
thf(fact_6917_relcompp__relcomp__eq,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( relcompp @ A @ B @ C
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R3 )
@ ^ [X4: B,Y4: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X4 @ Y4 ) @ S2 ) )
= ( ^ [X4: A,Y4: C] : ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Y4 ) @ ( relcomp @ A @ B @ C @ R3 @ S2 ) ) ) ) ).
% relcompp_relcomp_eq
thf(fact_6918_cut__apply,axiom,
! [B: $tType,A: $tType,X: A,A3: A,R2: set @ ( product_prod @ A @ A ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R2 )
=> ( ( cut @ A @ B @ F3 @ R2 @ A3 @ X )
= ( F3 @ X ) ) ) ).
% cut_apply
thf(fact_6919_cuts__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,R2: set @ ( product_prod @ A @ A ),X: A,G3: A > B] :
( ( ( cut @ A @ B @ F3 @ R2 @ X )
= ( cut @ A @ B @ G3 @ R2 @ X ) )
= ( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X ) @ R2 )
=> ( ( F3 @ Y4 )
= ( G3 @ Y4 ) ) ) ) ) ).
% cuts_eq
thf(fact_6920_relpowp_Osimps_I2_J,axiom,
! [A: $tType,N: nat,R2: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ R2 )
= ( relcompp @ A @ A @ A @ ( compow @ ( A > A > $o ) @ N @ R2 ) @ R2 ) ) ).
% relpowp.simps(2)
thf(fact_6921_relcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcomp @ A @ B @ C )
= ( ^ [R: set @ ( product_prod @ A @ B ),S6: set @ ( product_prod @ B @ C )] :
( collect @ ( product_prod @ A @ C )
@ ( product_case_prod @ A @ C @ $o
@ ( relcompp @ A @ B @ C
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R )
@ ^ [X4: B,Y4: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X4 @ Y4 ) @ S6 ) ) ) ) ) ) ).
% relcomp_def
thf(fact_6922_pred__option__parametric,axiom,
! [A: $tType,B: $tType,A6: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A6
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ A6 )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( pred_option @ A )
@ ( pred_option @ B ) ) ).
% pred_option_parametric
thf(fact_6923_option_Opred__transfer,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R2
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R2 )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
@ ( pred_option @ A )
@ ( pred_option @ B ) ) ).
% option.pred_transfer
thf(fact_6924_option_Opred__inject_I2_J,axiom,
! [A: $tType,P2: A > $o,A3: A] :
( ( pred_option @ A @ P2 @ ( some @ A @ A3 ) )
= ( P2 @ A3 ) ) ).
% option.pred_inject(2)
thf(fact_6925_option_Opred__mono,axiom,
! [A: $tType,P2: A > $o,Pa: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ P2 @ Pa )
=> ( ord_less_eq @ ( ( option @ A ) > $o ) @ ( pred_option @ A @ P2 ) @ ( pred_option @ A @ Pa ) ) ) ).
% option.pred_mono
thf(fact_6926_option_Opred__inject_I1_J,axiom,
! [A: $tType,P2: A > $o] : ( pred_option @ A @ P2 @ ( none @ A ) ) ).
% option.pred_inject(1)
thf(fact_6927_option_Opred__True,axiom,
! [A: $tType] :
( ( pred_option @ A
@ ^ [Uu3: A] : $true )
= ( ^ [Uu3: option @ A] : $true ) ) ).
% option.pred_True
thf(fact_6928_option_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: option @ A,Ya: option @ A,F3: A > B,G3: A > B] :
( ( X = Ya )
=> ( ( pred_option @ A
@ ^ [Z4: A] :
( ( F3 @ Z4 )
= ( G3 @ Z4 ) )
@ Ya )
=> ( ( map_option @ A @ B @ F3 @ X )
= ( map_option @ A @ B @ G3 @ Ya ) ) ) ) ).
% option.map_cong_pred
thf(fact_6929_option_Opred__cong,axiom,
! [A: $tType,X: option @ A,Ya: option @ A,P2: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( set_option @ A @ Ya ) )
=> ( ( P2 @ Z )
= ( Pa @ Z ) ) )
=> ( ( pred_option @ A @ P2 @ X )
= ( pred_option @ A @ Pa @ Ya ) ) ) ) ).
% option.pred_cong
thf(fact_6930_option_Opred__mono__strong,axiom,
! [A: $tType,P2: A > $o,X: option @ A,Pa: A > $o] :
( ( pred_option @ A @ P2 @ X )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( set_option @ A @ X ) )
=> ( ( P2 @ Z )
=> ( Pa @ Z ) ) )
=> ( pred_option @ A @ Pa @ X ) ) ) ).
% option.pred_mono_strong
thf(fact_6931_option_Opred__set,axiom,
! [A: $tType] :
( ( pred_option @ A )
= ( ^ [P4: A > $o,X4: option @ A] :
! [Y4: A] :
( ( member @ A @ Y4 @ ( set_option @ A @ X4 ) )
=> ( P4 @ Y4 ) ) ) ) ).
% option.pred_set
thf(fact_6932_option_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F3: A > B,X: option @ A] :
( ( pred_option @ B @ Q @ ( map_option @ A @ B @ F3 @ X ) )
= ( pred_option @ A @ ( comp @ B @ $o @ A @ Q @ F3 ) @ X ) ) ).
% option.pred_map
thf(fact_6933_above__def,axiom,
! [A: $tType] :
( ( order_above @ A )
= ( ^ [R: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R ) ) ) ) ).
% above_def
thf(fact_6934_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F12: T,X4: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X4 ) ) ) ) ).
% old.rec_unit_def
thf(fact_6935_length__removeAll__less,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_removeAll_less
thf(fact_6936_lcm__altdef__int,axiom,
( ( gcd_lcm @ int )
= ( ^ [A7: int,B5: int] : ( divide_divide @ int @ ( times_times @ int @ ( abs_abs @ int @ A7 ) @ ( abs_abs @ int @ B5 ) ) @ ( gcd_gcd @ int @ A7 @ B5 ) ) ) ) ).
% lcm_altdef_int
thf(fact_6937_lcm_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% lcm.bottom_right_bottom
thf(fact_6938_lcm_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% lcm.bottom_left_bottom
thf(fact_6939_lcm__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_lcm @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% lcm_eq_0_iff
thf(fact_6940_zero__eq__lcm__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( zero_zero @ A )
= ( gcd_lcm @ A @ A3 @ B2 ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_lcm_iff
thf(fact_6941_lcm__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( times_times @ A @ B2 @ A3 ) @ C3 )
= ( gcd_lcm @ A @ B2 @ C3 ) ) ) ) ).
% lcm_mult_unit1
thf(fact_6942_lcm__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B2 @ ( times_times @ A @ C3 @ A3 ) )
= ( gcd_lcm @ A @ B2 @ C3 ) ) ) ) ).
% lcm_mult_unit2
thf(fact_6943_length__removeAll__less__eq,axiom,
! [A: $tType,X: A,Xs2: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% length_removeAll_less_eq
thf(fact_6944_prod__gcd__lcm__int,axiom,
! [M2: int,N: int] :
( ( times_times @ int @ ( abs_abs @ int @ M2 ) @ ( abs_abs @ int @ N ) )
= ( times_times @ int @ ( gcd_gcd @ int @ M2 @ N ) @ ( gcd_lcm @ int @ M2 @ N ) ) ) ).
% prod_gcd_lcm_int
thf(fact_6945_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( ^ [A8: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A8 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A8 ) @ ( zero_zero @ A ) ) ) ) ) ).
% Lcm_fin.eq_fold
thf(fact_6946_Lcm__fin__def,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( bounde2362111253966948842tice_F @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ).
% Lcm_fin_def
thf(fact_6947_lcm__0__iff__nat,axiom,
! [M2: nat,N: nat] :
( ( ( gcd_lcm @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% lcm_0_iff_nat
thf(fact_6948_lcm__1__iff__nat,axiom,
! [M2: nat,N: nat] :
( ( ( gcd_lcm @ nat @ M2 @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% lcm_1_iff_nat
thf(fact_6949_Lcm__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( semiring_gcd_Lcm_fin @ A @ A6 )
= ( zero_zero @ A ) ) ) ) ).
% Lcm_fin.infinite
thf(fact_6950_prod__gcd__lcm__nat,axiom,
( ( times_times @ nat )
= ( ^ [M6: nat,N5: nat] : ( times_times @ nat @ ( gcd_gcd @ nat @ M6 @ N5 ) @ ( gcd_lcm @ nat @ M6 @ N5 ) ) ) ) ).
% prod_gcd_lcm_nat
thf(fact_6951_lcm__pos__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_lcm @ nat @ M2 @ N ) ) ) ) ).
% lcm_pos_nat
thf(fact_6952_lcm__code__integer,axiom,
( ( gcd_lcm @ code_integer )
= ( ^ [A7: code_integer,B5: code_integer] : ( divide_divide @ code_integer @ ( times_times @ code_integer @ ( abs_abs @ code_integer @ A7 ) @ ( abs_abs @ code_integer @ B5 ) ) @ ( gcd_gcd @ code_integer @ A7 @ B5 ) ) ) ) ).
% lcm_code_integer
thf(fact_6953_Lcm__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( semiring_gcd_Lcm_fin @ A @ A6 )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ A6 ) ) ) ) ).
% Lcm_fin_0_iff
thf(fact_6954_lcm__nat__def,axiom,
( ( gcd_lcm @ nat )
= ( ^ [X4: nat,Y4: nat] : ( divide_divide @ nat @ ( times_times @ nat @ X4 @ Y4 ) @ ( gcd_gcd @ nat @ X4 @ Y4 ) ) ) ) ).
% lcm_nat_def
thf(fact_6955_Lcm__eq__Max__nat,axiom,
! [M5: set @ nat] :
( ( finite_finite2 @ nat @ M5 )
=> ( ( M5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ! [M: nat,N3: nat] :
( ( member @ nat @ M @ M5 )
=> ( ( member @ nat @ N3 @ M5 )
=> ( member @ nat @ ( gcd_lcm @ nat @ M @ N3 ) @ M5 ) ) )
=> ( ( gcd_Lcm @ nat @ M5 )
= ( lattic643756798349783984er_Max @ nat @ M5 ) ) ) ) ) ) ).
% Lcm_eq_Max_nat
thf(fact_6956_gcd__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( gcd_gcd @ A @ A3 @ B2 )
= ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% gcd_lcm
thf(fact_6957_normalize__idem,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( normal6383669964737779283malize @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% normalize_idem
thf(fact_6958_normalize__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% normalize_eq_0_iff
thf(fact_6959_normalize__0,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% normalize_0
thf(fact_6960_lcm_Onormalize__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( normal6383669964737779283malize @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% lcm.normalize_bottom
thf(fact_6961_normalize__mult__normalize__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% normalize_mult_normalize_right
thf(fact_6962_normalize__mult__normalize__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% normalize_mult_normalize_left
thf(fact_6963_normalize__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% normalize_1
thf(fact_6964_normalize__dvd__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% normalize_dvd_iff
thf(fact_6965_dvd__normalize__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_normalize_iff
thf(fact_6966_coprime__normalize__right__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprime_normalize_right_iff
thf(fact_6967_coprime__normalize__left__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprime_normalize_left_iff
thf(fact_6968_Lcm__eq__0__I__nat,axiom,
! [A6: set @ nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ A6 )
=> ( ( gcd_Lcm @ nat @ A6 )
= ( zero_zero @ nat ) ) ) ).
% Lcm_eq_0_I_nat
thf(fact_6969_Lcm__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( top_top @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Lcm_UNIV
thf(fact_6970_gcd_Otop__right__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ A3 @ ( zero_zero @ A ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% gcd.top_right_normalize
thf(fact_6971_gcd_Otop__left__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ ( zero_zero @ A ) @ A3 )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% gcd.top_left_normalize
thf(fact_6972_Lcm__0__iff__nat,axiom,
! [A6: set @ nat] :
( ( finite_finite2 @ nat @ A6 )
=> ( ( ( gcd_Lcm @ nat @ A6 )
= ( zero_zero @ nat ) )
= ( member @ nat @ ( zero_zero @ nat ) @ A6 ) ) ) ).
% Lcm_0_iff_nat
thf(fact_6973_normalize__mult__unit__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ) ).
% normalize_mult_unit_right
thf(fact_6974_normalize__mult__unit__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% normalize_mult_unit_left
thf(fact_6975_gcd__mult__lcm,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% gcd_mult_lcm
thf(fact_6976_lcm__mult__gcd,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% lcm_mult_gcd
thf(fact_6977_lcm__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C3 ) @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( gcd_lcm @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ).
% lcm_mult_distrib'
thf(fact_6978_lcm__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_lcm @ A @ B2 @ A3 ) @ C3 ) ) ) ) ).
% lcm_mult_right
thf(fact_6979_lcm__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ).
% lcm_mult_left
thf(fact_6980_Lcm__coprime_H,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( ( finite_card @ A @ A6 )
!= ( zero_zero @ nat ) )
=> ( ! [A5: A,B4: A] :
( ( member @ A @ A5 @ A6 )
=> ( ( member @ A @ B4 @ A6 )
=> ( ( A5 != B4 )
=> ( algebr8660921524188924756oprime @ A @ A5 @ B4 ) ) ) )
=> ( ( gcd_Lcm @ A @ A6 )
= ( normal6383669964737779283malize @ A
@ ( groups7121269368397514597t_prod @ A @ A
@ ^ [X4: A] : X4
@ A6 ) ) ) ) ) ) ).
% Lcm_coprime'
thf(fact_6981_Lcm__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A,C3: A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( gcd_Lcm @ A @ ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ A6 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_Lcm @ A @ A6 ) ) ) ) ) ) ).
% Lcm_mult
thf(fact_6982_dvd__normalize__div,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ) ).
% dvd_normalize_div
thf(fact_6983_associated__iff__dvd,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
= ( ( dvd_dvd @ A @ A3 @ B2 )
& ( dvd_dvd @ A @ B2 @ A3 ) ) ) ) ).
% associated_iff_dvd
thf(fact_6984_associated__eqI,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ( ( normal6383669964737779283malize @ A @ B2 )
= B2 )
=> ( A3 = B2 ) ) ) ) ) ) ).
% associated_eqI
thf(fact_6985_associatedD2,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% associatedD2
thf(fact_6986_associatedD1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% associatedD1
thf(fact_6987_associatedI,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ) ).
% associatedI
thf(fact_6988_gcd__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ) ).
% gcd_mult_left
thf(fact_6989_gcd__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_gcd @ A @ B2 @ A3 ) @ C3 ) ) ) ) ).
% gcd_mult_right
thf(fact_6990_gcd__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C3 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( gcd_gcd @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ).
% gcd_mult_distrib'
thf(fact_6991_normalize__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% normalize_mult
thf(fact_6992_coprime__crossproduct,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,D3: A,B2: A,C3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ D3 )
=> ( ( algebr8660921524188924756oprime @ A @ B2 @ C3 )
=> ( ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ C3 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ B2 ) @ ( normal6383669964737779283malize @ A @ D3 ) ) )
= ( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
& ( ( normal6383669964737779283malize @ A @ C3 )
= ( normal6383669964737779283malize @ A @ D3 ) ) ) ) ) ) ) ).
% coprime_crossproduct
thf(fact_6993_Lcm__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( gcd_Lcm @ A @ A6 )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ A6 ) ) ) ) ).
% Lcm_0_iff
thf(fact_6994_Lcm__nat__infinite,axiom,
! [M5: set @ nat] :
( ~ ( finite_finite2 @ nat @ M5 )
=> ( ( gcd_Lcm @ nat @ M5 )
= ( zero_zero @ nat ) ) ) ).
% Lcm_nat_infinite
thf(fact_6995_Lcm__eq__0__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ A6 )
=> ( ( gcd_Lcm @ A @ A6 )
= ( zero_zero @ A ) ) ) ) ).
% Lcm_eq_0_I
thf(fact_6996_Lcm__0__iff_H,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( ( gcd_Lcm @ A @ A6 )
= ( zero_zero @ A ) )
= ( ~ ? [L2: A] :
( ( L2
!= ( zero_zero @ A ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( dvd_dvd @ A @ X4 @ L2 ) ) ) ) ) ) ).
% Lcm_0_iff'
thf(fact_6997_Lcm__no__multiple,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ! [M: A] :
( ( M
!= ( zero_zero @ A ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ~ ( dvd_dvd @ A @ X5 @ M ) ) )
=> ( ( gcd_Lcm @ A @ A6 )
= ( zero_zero @ A ) ) ) ) ).
% Lcm_no_multiple
thf(fact_6998_normalize__idem__imp__is__unit__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
= ( A3
= ( one_one @ A ) ) ) ) ) ).
% normalize_idem_imp_is_unit_iff
thf(fact_6999_is__unit__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% is_unit_normalize
thf(fact_7000_normalize__1__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( one_one @ A ) )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% normalize_1_iff
thf(fact_7001_associated__unit,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% associated_unit
thf(fact_7002_lcm__gcd__prod,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% lcm_gcd_prod
thf(fact_7003_Gcd__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [C3: A,A6: set @ A] :
( ( gcd_Gcd @ A @ ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ A6 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_Gcd @ A @ A6 ) ) ) ) ) ).
% Gcd_mult
thf(fact_7004_lcm__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( gcd_lcm @ A @ A3 @ B2 )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% lcm_coprime
thf(fact_7005_lcm__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_lcm @ A )
= ( ^ [A7: A,B5: A] : ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A7 @ B5 ) @ ( gcd_gcd @ A @ A7 @ B5 ) ) ) ) ) ) ).
% lcm_gcd
thf(fact_7006_Lcm__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A,B2: A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( semiring_gcd_Lcm_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B2 ) @ A6 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Lcm_fin @ A @ A6 ) ) ) ) ) ) ).
% Lcm_fin_mult
thf(fact_7007_Gcd__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A,B2: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( semiring_gcd_Gcd_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B2 ) @ A6 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Gcd_fin @ A @ A6 ) ) ) ) ) ) ).
% Gcd_fin_mult
thf(fact_7008_Lcm__nat__def,axiom,
( ( gcd_Lcm @ nat )
= ( ^ [M9: set @ nat] : ( if @ nat @ ( finite_finite2 @ nat @ M9 ) @ ( lattic5214292709420241887eutr_F @ nat @ ( gcd_lcm @ nat ) @ ( one_one @ nat ) @ M9 ) @ ( zero_zero @ nat ) ) ) ) ).
% Lcm_nat_def
thf(fact_7009_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde6485984586167503788ce_set @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% Lcm_fin.bounded_quasi_semilattice_set_axioms
thf(fact_7010_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde6485984586167503788ce_set @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% Gcd_fin.bounded_quasi_semilattice_set_axioms
thf(fact_7011_lcm_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde8507323023520639062attice @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% lcm.bounded_quasi_semilattice_axioms
thf(fact_7012_gcd_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde8507323023520639062attice @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% gcd.bounded_quasi_semilattice_axioms
thf(fact_7013_normalize__div,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ) ).
% normalize_div
thf(fact_7014_normalize__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% normalize_unit_factor
thf(fact_7015_unit__factor__simps_I1_J,axiom,
( ( unit_f5069060285200089521factor @ nat @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% unit_factor_simps(1)
thf(fact_7016_unit__factor__idem,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( unit_f5069060285200089521factor @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ).
% unit_factor_idem
thf(fact_7017_unit__factor__simps_I2_J,axiom,
! [N: nat] :
( ( unit_f5069060285200089521factor @ nat @ ( suc @ N ) )
= ( one_one @ nat ) ) ).
% unit_factor_simps(2)
thf(fact_7018_unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( unit_f5069060285200089521factor @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% unit_factor_eq_0_iff
thf(fact_7019_unit__factor__0,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% unit_factor_0
thf(fact_7020_unit__factor__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% unit_factor_1
thf(fact_7021_unit__factor__mult__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ A3 ) )
= A3 ) ) ).
% unit_factor_mult_normalize
thf(fact_7022_normalize__mult__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= A3 ) ) ).
% normalize_mult_unit_factor
thf(fact_7023_div__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ).
% div_normalize
thf(fact_7024_div__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% div_unit_factor
thf(fact_7025_inv__unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% inv_unit_factor_eq_0_iff
thf(fact_7026_mult__one__div__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) )
= ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% mult_one_div_unit_factor
thf(fact_7027_unit__factor__mult__unit__left,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% unit_factor_mult_unit_left
thf(fact_7028_unit__factor__mult__unit__right,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ B2 @ A3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ B2 ) @ A3 ) ) ) ) ).
% unit_factor_mult_unit_right
thf(fact_7029_unit__factor__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_lcm
thf(fact_7030_unit__factor__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% unit_factor_normalize
thf(fact_7031_lcm__mult__distrib,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [K2: A,A3: A,B2: A] :
( ( times_times @ A @ K2 @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ K2 @ A3 ) @ ( times_times @ A @ K2 @ B2 ) ) @ ( unit_f5069060285200089521factor @ A @ K2 ) ) ) ) ).
% lcm_mult_distrib
thf(fact_7032_mult__lcm__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ C3 )
= ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) @ ( unit_f5069060285200089521factor @ A @ C3 ) ) ) ) ).
% mult_lcm_right
thf(fact_7033_mult__lcm__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( times_times @ A @ C3 @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C3 ) @ ( gcd_lcm @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_lcm_left
thf(fact_7034_normalize__unit__factor__eqI,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( ( ( unit_f5069060285200089521factor @ A @ A3 )
= ( unit_f5069060285200089521factor @ A @ B2 ) )
=> ( A3 = B2 ) ) ) ) ).
% normalize_unit_factor_eqI
thf(fact_7035_dvd__unit__factor__div,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( unit_f5069060285200089521factor @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% dvd_unit_factor_div
thf(fact_7036_mult__gcd__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( times_times @ A @ C3 @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C3 ) @ ( gcd_gcd @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_gcd_left
thf(fact_7037_mult__gcd__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ C3 )
= ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) @ ( unit_f5069060285200089521factor @ A @ C3 ) ) ) ) ).
% mult_gcd_right
thf(fact_7038_gcd__mult__distrib,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [K2: A,A3: A,B2: A] :
( ( times_times @ A @ K2 @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ K2 @ A3 ) @ ( times_times @ A @ K2 @ B2 ) ) @ ( unit_f5069060285200089521factor @ A @ K2 ) ) ) ) ).
% gcd_mult_distrib
thf(fact_7039_is__unit__unit__factor,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ A3 )
= A3 ) ) ) ).
% is_unit_unit_factor
thf(fact_7040_unit__factor__nat__def,axiom,
( ( unit_f5069060285200089521factor @ nat )
= ( ^ [N5: nat] :
( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( one_one @ nat ) ) ) ) ).
% unit_factor_nat_def
thf(fact_7041_unit__factor__dvd,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ B2 ) ) ) ).
% unit_factor_dvd
thf(fact_7042_unit__factor__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A3: A,B2: A] :
( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% unit_factor_mult
thf(fact_7043_unit__factor__self,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ A3 ) ) ).
% unit_factor_self
thf(fact_7044_unit__factor__is__unit,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( one_one @ A ) ) ) ) ).
% unit_factor_is_unit
thf(fact_7045_unit__factor__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_gcd
thf(fact_7046_coprime__crossproduct_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [B2: A,D3: A,A3: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( unit_f5069060285200089521factor @ A @ B2 )
= ( unit_f5069060285200089521factor @ A @ D3 ) )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( algebr8660921524188924756oprime @ A @ C3 @ D3 )
=> ( ( ( times_times @ A @ A3 @ D3 )
= ( times_times @ A @ B2 @ C3 ) )
= ( ( A3 = C3 )
& ( B2 = D3 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
thf(fact_7047_unit__factor__Lcm,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( ( ( gcd_Lcm @ A @ A6 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A6 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Lcm @ A @ A6 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A6 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Lcm
thf(fact_7048_unit__factor__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( ( ( gcd_Gcd @ A @ A6 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A6 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Gcd @ A @ A6 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A6 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Gcd
thf(fact_7049_normalize__idem__imp__unit__factor__eq,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ( unit_f5069060285200089521factor @ A @ A3 )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ) ).
% normalize_idem_imp_unit_factor_eq
thf(fact_7050_unit__factor__Lcm__fin,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Lcm_fin @ A @ A6 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( semiring_gcd_Lcm_fin @ A @ A6 )
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_factor_Lcm_fin
thf(fact_7051_unit__factor__Gcd__fin,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Gcd_fin @ A @ A6 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( semiring_gcd_Gcd_fin @ A @ A6 )
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_factor_Gcd_fin
thf(fact_7052_map__comp__None__iff,axiom,
! [A: $tType,C: $tType,B: $tType,M12: B > ( option @ A ),M23: C > ( option @ B ),K2: C] :
( ( ( map_comp @ B @ A @ C @ M12 @ M23 @ K2 )
= ( none @ A ) )
= ( ( ( M23 @ K2 )
= ( none @ B ) )
| ? [K10: B] :
( ( ( M23 @ K2 )
= ( some @ B @ K10 ) )
& ( ( M12 @ K10 )
= ( none @ A ) ) ) ) ) ).
% map_comp_None_iff
thf(fact_7053_map__le__imp__upd__le,axiom,
! [A: $tType,B: $tType,M12: A > ( option @ B ),M23: A > ( option @ B ),X: A,Y: B] :
( ( map_le @ A @ B @ M12 @ M23 )
=> ( map_le @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M12 @ X @ ( none @ B ) ) @ ( fun_upd @ A @ ( option @ B ) @ M23 @ X @ ( some @ B @ Y ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_7054_map__comp__simps_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,M23: B > ( option @ A ),K2: B,K7: A,M12: A > ( option @ C )] :
( ( ( M23 @ K2 )
= ( some @ A @ K7 ) )
=> ( ( map_comp @ A @ C @ B @ M12 @ M23 @ K2 )
= ( M12 @ K7 ) ) ) ).
% map_comp_simps(2)
thf(fact_7055_map__comp__Some__iff,axiom,
! [A: $tType,C: $tType,B: $tType,M12: B > ( option @ A ),M23: C > ( option @ B ),K2: C,V3: A] :
( ( ( map_comp @ B @ A @ C @ M12 @ M23 @ K2 )
= ( some @ A @ V3 ) )
= ( ? [K10: B] :
( ( ( M23 @ K2 )
= ( some @ B @ K10 ) )
& ( ( M12 @ K10 )
= ( some @ A @ V3 ) ) ) ) ) ).
% map_comp_Some_iff
thf(fact_7056_is__num_Ocases,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A] :
( ( neg_numeral_is_num @ A @ A3 )
=> ( ( A3
!= ( one_one @ A ) )
=> ( ! [X3: A] :
( ( A3
= ( uminus_uminus @ A @ X3 ) )
=> ~ ( neg_numeral_is_num @ A @ X3 ) )
=> ~ ! [X3: A,Y3: A] :
( ( A3
= ( plus_plus @ A @ X3 @ Y3 ) )
=> ( ( neg_numeral_is_num @ A @ X3 )
=> ~ ( neg_numeral_is_num @ A @ Y3 ) ) ) ) ) ) ) ).
% is_num.cases
thf(fact_7057_is__num_Osimps,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_is_num @ A )
= ( ^ [A7: A] :
( ( A7
= ( one_one @ A ) )
| ? [X4: A] :
( ( A7
= ( uminus_uminus @ A @ X4 ) )
& ( neg_numeral_is_num @ A @ X4 ) )
| ? [X4: A,Y4: A] :
( ( A7
= ( plus_plus @ A @ X4 @ Y4 ) )
& ( neg_numeral_is_num @ A @ X4 )
& ( neg_numeral_is_num @ A @ Y4 ) ) ) ) ) ) ).
% is_num.simps
thf(fact_7058_is__num__normalize_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( neg_numeral_is_num @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% is_num_normalize(5)
thf(fact_7059_is__num__normalize_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( neg_numeral_is_num @ A @ ( one_one @ A ) ) ) ).
% is_num_normalize(4)
thf(fact_7060_is__num__normalize_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A,Y: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( ( neg_numeral_is_num @ A @ Y )
=> ( neg_numeral_is_num @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ) ).
% is_num_normalize(6)
thf(fact_7061_is__num__add__commute,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A,Y: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( ( neg_numeral_is_num @ A @ Y )
=> ( ( plus_plus @ A @ X @ Y )
= ( plus_plus @ A @ Y @ X ) ) ) ) ) ).
% is_num_add_commute
thf(fact_7062_is__num__add__left__commute,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( ( neg_numeral_is_num @ A @ Y )
=> ( ( plus_plus @ A @ X @ ( plus_plus @ A @ Y @ Z3 ) )
= ( plus_plus @ A @ Y @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ) ) ).
% is_num_add_left_commute
thf(fact_7063_is__num__numeral,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] : ( neg_numeral_is_num @ A @ ( numeral_numeral @ A @ K2 ) ) ) ).
% is_num_numeral
thf(fact_7064_times__num__def,axiom,
( ( times_times @ num )
= ( ^ [M6: num,N5: num] : ( num_of_nat @ ( times_times @ nat @ ( nat_of_num @ M6 ) @ ( nat_of_num @ N5 ) ) ) ) ) ).
% times_num_def
thf(fact_7065_arg__max__nat__lemma,axiom,
! [A: $tType,P2: A > $o,K2: A,F3: A > nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less @ nat @ ( F3 @ Y3 ) @ B2 ) )
=> ( ( P2 @ ( lattices_ord_arg_max @ A @ nat @ F3 @ P2 ) )
& ! [Y6: A] :
( ( P2 @ Y6 )
=> ( ord_less_eq @ nat @ ( F3 @ Y6 ) @ ( F3 @ ( lattices_ord_arg_max @ A @ nat @ F3 @ P2 ) ) ) ) ) ) ) ).
% arg_max_nat_lemma
thf(fact_7066_nat__of__num__code_I2_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit0 @ N ) )
= ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% nat_of_num_code(2)
thf(fact_7067_less__eq__num__def,axiom,
( ( ord_less_eq @ num )
= ( ^ [M6: num,N5: num] : ( ord_less_eq @ nat @ ( nat_of_num @ M6 ) @ ( nat_of_num @ N5 ) ) ) ) ).
% less_eq_num_def
thf(fact_7068_less__num__def,axiom,
( ( ord_less @ num )
= ( ^ [M6: num,N5: num] : ( ord_less @ nat @ ( nat_of_num @ M6 ) @ ( nat_of_num @ N5 ) ) ) ) ).
% less_num_def
thf(fact_7069_nat__of__num__inc,axiom,
! [X: num] :
( ( nat_of_num @ ( inc @ X ) )
= ( suc @ ( nat_of_num @ X ) ) ) ).
% nat_of_num_inc
thf(fact_7070_nat__of__num__mult,axiom,
! [X: num,Y: num] :
( ( nat_of_num @ ( times_times @ num @ X @ Y ) )
= ( times_times @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% nat_of_num_mult
thf(fact_7071_nat__of__num__numeral,axiom,
( nat_of_num
= ( numeral_numeral @ nat ) ) ).
% nat_of_num_numeral
thf(fact_7072_num__eq__iff,axiom,
( ( ^ [Y5: num,Z2: num] : Y5 = Z2 )
= ( ^ [X4: num,Y4: num] :
( ( nat_of_num @ X4 )
= ( nat_of_num @ Y4 ) ) ) ) ).
% num_eq_iff
thf(fact_7073_nat__of__num__inverse,axiom,
! [X: num] :
( ( num_of_nat @ ( nat_of_num @ X ) )
= X ) ).
% nat_of_num_inverse
thf(fact_7074_nat__of__num_Osimps_I2_J,axiom,
! [X: num] :
( ( nat_of_num @ ( bit0 @ X ) )
= ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% nat_of_num.simps(2)
thf(fact_7075_nat__of__num__pos,axiom,
! [X: num] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat_of_num @ X ) ) ).
% nat_of_num_pos
thf(fact_7076_nat__of__num__neq__0,axiom,
! [X: num] :
( ( nat_of_num @ X )
!= ( zero_zero @ nat ) ) ).
% nat_of_num_neq_0
thf(fact_7077_nat__of__num__code_I1_J,axiom,
( ( nat_of_num @ one2 )
= ( one_one @ nat ) ) ).
% nat_of_num_code(1)
thf(fact_7078_nat__of__num__add,axiom,
! [X: num,Y: num] :
( ( nat_of_num @ ( plus_plus @ num @ X @ Y ) )
= ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% nat_of_num_add
thf(fact_7079_nat__of__num__sqr,axiom,
! [X: num] :
( ( nat_of_num @ ( sqr @ X ) )
= ( times_times @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% nat_of_num_sqr
thf(fact_7080_arg__max__equality,axiom,
! [A: $tType,C: $tType] :
( ( order @ A )
=> ! [P2: C > $o,K2: C,F3: C > A] :
( ( P2 @ K2 )
=> ( ! [X3: C] :
( ( P2 @ X3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ K2 ) ) )
=> ( ( F3 @ ( lattices_ord_arg_max @ C @ A @ F3 @ P2 ) )
= ( F3 @ K2 ) ) ) ) ) ).
% arg_max_equality
thf(fact_7081_nat__of__num_Osimps_I1_J,axiom,
( ( nat_of_num @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_of_num.simps(1)
thf(fact_7082_nat__of__num_Osimps_I3_J,axiom,
! [X: num] :
( ( nat_of_num @ ( bit1 @ X ) )
= ( suc @ ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).
% nat_of_num.simps(3)
thf(fact_7083_num__of__nat__inverse,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nat_of_num @ ( num_of_nat @ N ) )
= N ) ) ).
% num_of_nat_inverse
thf(fact_7084_nat__of__num__code_I3_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit1 @ N ) )
= ( suc @ ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% nat_of_num_code(3)
thf(fact_7085_plus__num__def,axiom,
( ( plus_plus @ num )
= ( ^ [M6: num,N5: num] : ( num_of_nat @ ( plus_plus @ nat @ ( nat_of_num @ M6 ) @ ( nat_of_num @ N5 ) ) ) ) ) ).
% plus_num_def
thf(fact_7086_arg__max__nat__le,axiom,
! [A: $tType,P2: A > $o,X: A,F3: A > nat,B2: nat] :
( ( P2 @ X )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less @ nat @ ( F3 @ Y3 ) @ B2 ) )
=> ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ ( lattices_ord_arg_max @ A @ nat @ F3 @ P2 ) ) ) ) ) ).
% arg_max_nat_le
thf(fact_7087_trans__join,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y4: A,Y16: A] :
! [Z4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ Z4 @ R )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y24: A,Aa4: A] :
( ( Y16 = Y24 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Aa4 ) @ R ) )
@ Z4 ) )
@ X4 ) ) ) ) ).
% trans_join
thf(fact_7088_Gr__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr @ A @ B )
= ( ^ [A8: set @ A,F4: A > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A7: A] :
( ( Uu3
= ( product_Pair @ A @ B @ A7 @ ( F4 @ A7 ) ) )
& ( member @ A @ A7 @ A8 ) ) ) ) ) ).
% Gr_def
thf(fact_7089_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [X4: A,Y4: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z4 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z4 ) @ R ) ) ) ) ) ).
% trans_def
thf(fact_7090_transI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A,Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z ) @ R3 ) ) )
=> ( trans @ A @ R3 ) ) ).
% transI
thf(fact_7091_transE,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A,Z3: A] :
( ( trans @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ R3 ) ) ) ) ).
% transE
thf(fact_7092_transD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A,Z3: A] :
( ( trans @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ R3 ) ) ) ) ).
% transD
thf(fact_7093_GrD1,axiom,
! [B: $tType,A: $tType,X: A,Fx: B,A6: set @ A,F3: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A6 @ F3 ) )
=> ( member @ A @ X @ A6 ) ) ).
% GrD1
thf(fact_7094_GrD2,axiom,
! [A: $tType,B: $tType,X: A,Fx: B,A6: set @ A,F3: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A6 @ F3 ) )
=> ( ( F3 @ X )
= Fx ) ) ).
% GrD2
thf(fact_7095_natLeq__trans,axiom,
trans @ nat @ bNF_Ca8665028551170535155natLeq ).
% natLeq_trans
thf(fact_7096_transp__trans__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transp @ A
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) )
= ( trans @ A @ R3 ) ) ).
% transp_trans_eq
thf(fact_7097_trans__singleton,axiom,
! [A: $tType,A3: A] : ( trans @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trans_singleton
thf(fact_7098_wf__finite__segments,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R3 )
=> ( ( trans @ A @ R3 )
=> ( ! [X3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X3 ) @ R3 ) ) )
=> ( wf @ A @ R3 ) ) ) ) ).
% wf_finite_segments
thf(fact_7099_underS__incr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( trans @ A @ R3 )
=> ( ( antisym @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( order_underS @ A @ R3 @ B2 ) ) ) ) ) ).
% underS_incr
thf(fact_7100_natLeq__antisym,axiom,
antisym @ nat @ bNF_Ca8665028551170535155natLeq ).
% natLeq_antisym
thf(fact_7101_irreflI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R2 )
=> ( irrefl @ A @ R2 ) ) ).
% irreflI
thf(fact_7102_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [A7: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A7 ) @ R ) ) ) ).
% irrefl_def
thf(fact_7103_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R )
=> ( X4 = Y4 ) ) ) ) ) ).
% antisym_def
thf(fact_7104_antisymI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 )
=> ( X3 = Y3 ) ) )
=> ( antisym @ A @ R3 ) ) ).
% antisymI
thf(fact_7105_antisymD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( antisym @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 )
=> ( A3 = B2 ) ) ) ) ).
% antisymD
thf(fact_7106_irreflp__irrefl__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irreflp @ A
@ ^ [A7: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R2 ) )
= ( irrefl @ A @ R2 ) ) ).
% irreflp_irrefl_eq
thf(fact_7107_antisymp__antisym__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( antisymp @ A
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) )
= ( antisym @ A @ R3 ) ) ).
% antisymp_antisym_eq
thf(fact_7108_prod_H__def,axiom,
! [C: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ C @ A )
= ( groups_comm_monoid_G @ A @ C @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod'_def
thf(fact_7109_prod_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups4802862169904069756st_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod.comm_monoid_list_set_axioms
thf(fact_7110_sum_H__def,axiom,
! [C: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ C @ A )
= ( groups_comm_monoid_G @ A @ C @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% sum'_def
thf(fact_7111_sum_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups4802862169904069756st_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum.comm_monoid_list_set_axioms
thf(fact_7112_VEBT__internal_Ooption__comp__shift_Opelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: option @ A,Xb: option @ A] :
( ~ ( vEBT_V6923181176774028177_shift @ A @ X @ Xa2 @ Xb )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ Xa2 @ Xb ) ) )
=> ( ( ( Xa2
= ( none @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Xb ) ) ) )
=> ( ! [V2: A] :
( ( Xa2
= ( some @ A @ V2 ) )
=> ( ( Xb
= ( none @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V2 ) @ ( none @ A ) ) ) ) ) )
=> ~ ! [X3: A] :
( ( Xa2
= ( some @ A @ X3 ) )
=> ! [Y3: A] :
( ( Xb
= ( some @ A @ Y3 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X3 ) @ ( some @ A @ Y3 ) ) ) )
=> ( X @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.pelims(3)
thf(fact_7113_VEBT__internal_Ooption__comp__shift_Opelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: option @ A,Xb: option @ A,Y: $o] :
( ( ( vEBT_V6923181176774028177_shift @ A @ X @ Xa2 @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ Xa2 @ Xb ) ) )
=> ( ( ( Xa2
= ( none @ A ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Xb ) ) ) ) )
=> ( ! [V2: A] :
( ( Xa2
= ( some @ A @ V2 ) )
=> ( ( Xb
= ( none @ A ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V2 ) @ ( none @ A ) ) ) ) ) ) )
=> ~ ! [X3: A] :
( ( Xa2
= ( some @ A @ X3 ) )
=> ! [Y3: A] :
( ( Xb
= ( some @ A @ Y3 ) )
=> ( ( Y
= ( X @ X3 @ Y3 ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X3 ) @ ( some @ A @ Y3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.pelims(1)
thf(fact_7114_VEBT__internal_Ooption__comp__shift_Oelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: option @ A,Xb: option @ A] :
( ( vEBT_V6923181176774028177_shift @ A @ X @ Xa2 @ Xb )
=> ~ ! [X3: A] :
( ( Xa2
= ( some @ A @ X3 ) )
=> ! [Y3: A] :
( ( Xb
= ( some @ A @ Y3 ) )
=> ~ ( X @ X3 @ Y3 ) ) ) ) ).
% VEBT_internal.option_comp_shift.elims(2)
thf(fact_7115_VEBT__internal_Ooption__comp__shift_Osimps_I3_J,axiom,
! [A: $tType,F3: A > A > $o,X: A,Y: A] :
( ( vEBT_V6923181176774028177_shift @ A @ F3 @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( F3 @ X @ Y ) ) ).
% VEBT_internal.option_comp_shift.simps(3)
thf(fact_7116_VEBT__internal_Olesseq_Oelims_I3_J,axiom,
! [X: option @ nat,Xa2: option @ nat] :
( ~ ( vEBT_VEBT_lesseq @ X @ Xa2 )
=> ~ ( vEBT_V6923181176774028177_shift @ nat @ ( ord_less_eq @ nat ) @ X @ Xa2 ) ) ).
% VEBT_internal.lesseq.elims(3)
thf(fact_7117_VEBT__internal_Olesseq_Oelims_I2_J,axiom,
! [X: option @ nat,Xa2: option @ nat] :
( ( vEBT_VEBT_lesseq @ X @ Xa2 )
=> ( vEBT_V6923181176774028177_shift @ nat @ ( ord_less_eq @ nat ) @ X @ Xa2 ) ) ).
% VEBT_internal.lesseq.elims(2)
thf(fact_7118_VEBT__internal_Olesseq_Oelims_I1_J,axiom,
! [X: option @ nat,Xa2: option @ nat,Y: $o] :
( ( ( vEBT_VEBT_lesseq @ X @ Xa2 )
= Y )
=> ( Y
= ( vEBT_V6923181176774028177_shift @ nat @ ( ord_less_eq @ nat ) @ X @ Xa2 ) ) ) ).
% VEBT_internal.lesseq.elims(1)
thf(fact_7119_VEBT__internal_Olesseq_Osimps,axiom,
( vEBT_VEBT_lesseq
= ( vEBT_V6923181176774028177_shift @ nat @ ( ord_less_eq @ nat ) ) ) ).
% VEBT_internal.lesseq.simps
thf(fact_7120_VEBT__internal_Ooption__comp__shift_Opelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: option @ A,Xb: option @ A] :
( ( vEBT_V6923181176774028177_shift @ A @ X @ Xa2 @ Xb )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ Xa2 @ Xb ) ) )
=> ~ ! [X3: A] :
( ( Xa2
= ( some @ A @ X3 ) )
=> ! [Y3: A] :
( ( Xb
= ( some @ A @ Y3 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) ) @ ( vEBT_V4810408830578336424ft_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ X @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X3 ) @ ( some @ A @ Y3 ) ) ) )
=> ~ ( X @ X3 @ Y3 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.pelims(2)
thf(fact_7121_VEBT__internal_Ooption__comp__shift_Oelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: option @ A,Xb: option @ A] :
( ~ ( vEBT_V6923181176774028177_shift @ A @ X @ Xa2 @ Xb )
=> ( ( Xa2
!= ( none @ A ) )
=> ( ( ? [V2: A] :
( Xa2
= ( some @ A @ V2 ) )
=> ( Xb
!= ( none @ A ) ) )
=> ~ ! [X3: A] :
( ( Xa2
= ( some @ A @ X3 ) )
=> ! [Y3: A] :
( ( Xb
= ( some @ A @ Y3 ) )
=> ( X @ X3 @ Y3 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.elims(3)
thf(fact_7122_VEBT__internal_Ooption__comp__shift_Oelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: option @ A,Xb: option @ A,Y: $o] :
( ( ( vEBT_V6923181176774028177_shift @ A @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2
= ( none @ A ) )
=> Y )
=> ( ( ? [V2: A] :
( Xa2
= ( some @ A @ V2 ) )
=> ( ( Xb
= ( none @ A ) )
=> Y ) )
=> ~ ! [X3: A] :
( ( Xa2
= ( some @ A @ X3 ) )
=> ! [Y3: A] :
( ( Xb
= ( some @ A @ Y3 ) )
=> ( Y
= ( ~ ( X @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.elims(1)
thf(fact_7123_VEBT__internal_Ooption__comp__shift_Osimps_I2_J,axiom,
! [A: $tType,Uw2: A > A > $o,V3: A] :
~ ( vEBT_V6923181176774028177_shift @ A @ Uw2 @ ( some @ A @ V3 ) @ ( none @ A ) ) ).
% VEBT_internal.option_comp_shift.simps(2)
thf(fact_7124_fold__atLeastAtMost__nat,axiom,
! [A: $tType,F3: nat > A > A,A3: nat,B2: nat,Acc3: A] :
( ( finite6289374366891150609ommute @ nat @ A @ F3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A3 @ B2 @ Acc3 )
= ( finite_fold @ nat @ A @ F3 @ Acc3 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) ) ) ) ).
% fold_atLeastAtMost_nat
thf(fact_7125_max_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice_order @ A @ ( ord_max @ A )
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ) ).
% max.semilattice_order_axioms
thf(fact_7126_inf_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semilattice_order @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% inf.semilattice_order_axioms
thf(fact_7127_min_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice_order @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% min.semilattice_order_axioms
thf(fact_7128_sup_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semilattice_order @ A @ ( sup_sup @ A )
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ) ).
% sup.semilattice_order_axioms
thf(fact_7129_under__incr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( trans @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R3 @ A3 ) @ ( order_under @ A @ R3 @ B2 ) ) ) ) ).
% under_incr
thf(fact_7130_bdd__below__primitive__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( condit16957441358409770ng_bdd @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ).
% bdd_below_primitive_def
thf(fact_7131_under__def,axiom,
! [A: $tType] :
( ( order_under @ A )
= ( ^ [R: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R ) ) ) ) ).
% under_def
thf(fact_7132_bdd__above__primitive__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit941137186595557371_above @ A )
= ( condit16957441358409770ng_bdd @ A @ ( ord_less_eq @ A ) ) ) ) ).
% bdd_above_primitive_def
thf(fact_7133_has__vector__derivative__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: real > A,F6: A,X: real,S2: set @ real,G3: real > A,G6: A] :
( ( has_ve8173657378732805170vative @ A @ F3 @ F6 @ ( topolo174197925503356063within @ real @ X @ S2 ) )
=> ( ( has_ve8173657378732805170vative @ A @ G3 @ G6 @ ( topolo174197925503356063within @ real @ X @ S2 ) )
=> ( has_ve8173657378732805170vative @ A
@ ^ [X4: real] : ( times_times @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F3 @ X ) @ G6 ) @ ( times_times @ A @ F6 @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ real @ X @ S2 ) ) ) ) ) ).
% has_vector_derivative_mult
thf(fact_7134_integer__of__nat__0,axiom,
( ( code_integer_of_nat @ ( zero_zero @ nat ) )
= ( zero_zero @ code_integer ) ) ).
% integer_of_nat_0
thf(fact_7135_has__vector__derivative__mult__right,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: real > A,X: A,F5: filter @ real,A3: A] :
( ( has_ve8173657378732805170vative @ A @ F3 @ X @ F5 )
=> ( has_ve8173657378732805170vative @ A
@ ^ [X4: real] : ( times_times @ A @ A3 @ ( F3 @ X4 ) )
@ ( times_times @ A @ A3 @ X )
@ F5 ) ) ) ).
% has_vector_derivative_mult_right
thf(fact_7136_has__vector__derivative__mult__left,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: real > A,X: A,F5: filter @ real,A3: A] :
( ( has_ve8173657378732805170vative @ A @ F3 @ X @ F5 )
=> ( has_ve8173657378732805170vative @ A
@ ^ [X4: real] : ( times_times @ A @ ( F3 @ X4 ) @ A3 )
@ ( times_times @ A @ X @ A3 )
@ F5 ) ) ) ).
% has_vector_derivative_mult_left
thf(fact_7137_has__vector__derivative__const,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C3: A,Net: filter @ real] :
( has_ve8173657378732805170vative @ A
@ ^ [X4: real] : C3
@ ( zero_zero @ A )
@ Net ) ) ).
% has_vector_derivative_const
thf(fact_7138_equiv__class__nondisjoint,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,A3: A,B2: A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_7139_bdd__below_Opreordering__bdd__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit622319405099724424ng_bdd @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_7140_equiv__class__eq__iff,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
= ( ( ( image @ A @ A @ R3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( member @ A @ X @ A6 )
& ( member @ A @ Y @ A6 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_7141_eq__equiv__class__iff,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( member @ A @ X @ A6 )
=> ( ( member @ A @ Y @ A6 )
=> ( ( ( image @ A @ A @ R3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_7142_equiv__class__eq,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_7143_eq__equiv__class,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A,A6: set @ A] :
( ( ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( member @ A @ B2 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ).
% eq_equiv_class
thf(fact_7144_bdd__above_Opreordering__bdd__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit622319405099724424ng_bdd @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% bdd_above.preordering_bdd_axioms
thf(fact_7145_subset__equiv__class,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),B2: A,A3: A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ A @ B2 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ).
% subset_equiv_class
thf(fact_7146_equiv__class__subset,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_7147_proj__iff,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 )
=> ( ( ( equiv_proj @ A @ A @ R3 @ X )
= ( equiv_proj @ A @ A @ R3 @ Y ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% proj_iff
thf(fact_7148_finite__refines__card__le,axiom,
! [A: $tType,A6: set @ A,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A6 @ R2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 )
=> ( ( equiv_equiv @ A @ A6 @ R2 )
=> ( ( 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 @ R2 ) ) ) ) ) ) ) ).
% finite_refines_card_le
thf(fact_7149_in__quotient__imp__closed,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),X8: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A6 @ R3 ) )
=> ( ( member @ A @ X @ X8 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ A @ Y @ X8 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_7150_quotient__eq__iff,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),X8: set @ A,Y7: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A6 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y7 @ ( equiv_quotient @ A @ A6 @ R3 ) )
=> ( ( member @ A @ X @ X8 )
=> ( ( member @ A @ Y @ Y7 )
=> ( ( X8 = Y7 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_7151_quotient__eqI,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),X8: set @ A,Y7: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A6 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y7 @ ( equiv_quotient @ A @ A6 @ R3 ) )
=> ( ( member @ A @ X @ X8 )
=> ( ( member @ A @ Y @ Y7 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( X8 = Y7 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_7152_eq__equiv__class__iff2,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( member @ A @ X @ A6 )
=> ( ( member @ A @ Y @ A6 )
=> ( ( ( equiv_quotient @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
= ( equiv_quotient @ A @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) @ R3 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_7153_in__quotient__imp__in__rel,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),X8: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A6 @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_7154_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),F3: A > ( set @ B ),X8: set @ A,Y7: set @ A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R3 @ F3 )
=> ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ X8 ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ Y7 ) ) )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A6 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y7 @ ( equiv_quotient @ A @ A6 @ R3 ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ( member @ A @ Y3 @ A6 )
=> ( ( ( F3 @ X3 )
= ( F3 @ Y3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) )
=> ( X8 = Y7 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_7155_disjnt__equiv__class,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( disjnt @ A @ ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_7156_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C5: set @ A,A6: set @ B,B6: set @ B] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ C5
@ ^ [Uu3: A] : A6 )
@ ( product_Sigma @ A @ B @ C5
@ ^ [Uu3: A] : B6 ) )
= ( ( C5
= ( bot_bot @ ( set @ A ) ) )
| ( disjnt @ B @ A6 @ B6 ) ) ) ).
% disjnt_Times1_iff
thf(fact_7157_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A6: set @ A,C5: set @ B,B6: set @ A] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu3: A] : C5 ) )
= ( ( C5
= ( bot_bot @ ( set @ B ) ) )
| ( disjnt @ A @ A6 @ B6 ) ) ) ).
% disjnt_Times2_iff
thf(fact_7158_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A6: set @ A,C5: A > ( set @ B ),B6: set @ A] :
( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A6 @ C5 ) @ ( product_Sigma @ A @ B @ B6 @ C5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
=> ( ( C5 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) )
| ( disjnt @ A @ A6 @ B6 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_7159_congruentD,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F3: A > B,Y: A,Z3: A] :
( ( equiv_congruent @ A @ B @ R3 @ F3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ R3 )
=> ( ( F3 @ Y )
= ( F3 @ Z3 ) ) ) ) ).
% congruentD
thf(fact_7160_congruentI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F3: A > B] :
( ! [Y3: A,Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R3 )
=> ( ( F3 @ Y3 )
= ( F3 @ Z ) ) )
=> ( equiv_congruent @ A @ B @ R3 @ F3 ) ) ).
% congruentI
thf(fact_7161_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),F3: A > A > B] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ! [Y3: A,Z: A] :
( ( member @ A @ Y3 @ A6 )
=> ( ( member @ A @ Z @ A6 )
=> ( ( F3 @ Y3 @ Z )
= ( F3 @ Z @ Y3 ) ) ) )
=> ( ! [Y3: A,Z: A,W: A] :
( ( member @ A @ W @ A6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R3 )
=> ( ( F3 @ W @ Y3 )
= ( F3 @ W @ Z ) ) ) )
=> ( equiv_congruent2 @ A @ A @ B @ R3 @ R3 @ F3 ) ) ) ) ).
% congruent2_commuteI
thf(fact_7162_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A18: set @ A,R1: set @ ( product_prod @ A @ A ),A26: set @ B,R22: set @ ( product_prod @ B @ B ),F3: A > B > C] :
( ( equiv_equiv @ A @ A18 @ R1 )
=> ( ( equiv_equiv @ B @ A26 @ R22 )
=> ( ! [Y3: A,Z: A,W: B] :
( ( member @ B @ W @ A26 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R1 )
=> ( ( F3 @ Y3 @ W )
= ( F3 @ Z @ W ) ) ) )
=> ( ! [Y3: B,Z: B,W: A] :
( ( member @ A @ W @ A18 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y3 @ Z ) @ R22 )
=> ( ( F3 @ W @ Y3 )
= ( F3 @ W @ Z ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F3 ) ) ) ) ) ).
% congruent2I
thf(fact_7163_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F3: A > B > C] :
( ! [Y15: A,Z12: A,Y23: B,Z23: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y15 @ Z12 ) @ R1 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y23 @ Z23 ) @ R22 )
=> ( ( F3 @ Y15 @ Y23 )
= ( F3 @ Z12 @ Z23 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F3 ) ) ).
% congruent2I'
thf(fact_7164_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F3: A > B > C,Y1: A,Z1: A,Y2: B,Z22: B] :
( ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z1 ) @ R1 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z22 ) @ R22 )
=> ( ( F3 @ Y1 @ Y2 )
= ( F3 @ Z1 @ Z22 ) ) ) ) ) ).
% congruent2D
thf(fact_7165_equivp__equiv,axiom,
! [A: $tType,A6: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ ( top_top @ ( set @ A ) ) @ A6 )
= ( equiv_equivp @ A
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ A6 ) ) ) ).
% equivp_equiv
thf(fact_7166_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P4: A > B > $o,Q6: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q6 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_7167_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P4: C > A > $o,Q6: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q6 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_7168_reflp__refl__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( reflp @ A
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) )
= ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R3 ) ) ).
% reflp_refl_eq
thf(fact_7169_option_Orel__reflp,axiom,
! [A: $tType,R2: A > A > $o] :
( ( reflp @ A @ R2 )
=> ( reflp @ ( option @ A ) @ ( rel_option @ A @ A @ R2 ) ) ) ).
% option.rel_reflp
thf(fact_7170_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_7171_natural__decr,axiom,
! [N: code_natural] :
( ( N
!= ( zero_zero @ code_natural ) )
=> ( ord_less @ nat @ ( minus_minus @ nat @ ( code_nat_of_natural @ N ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( code_nat_of_natural @ N ) ) ) ).
% natural_decr
thf(fact_7172_times__natural_Orep__eq,axiom,
! [X: code_natural,Xa2: code_natural] :
( ( code_nat_of_natural @ ( times_times @ code_natural @ X @ Xa2 ) )
= ( times_times @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa2 ) ) ) ).
% times_natural.rep_eq
thf(fact_7173_zero__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% zero_natural.rep_eq
thf(fact_7174_less__eq__natural_Orep__eq,axiom,
( ( ord_less_eq @ code_natural )
= ( ^ [X4: code_natural,Xa3: code_natural] : ( ord_less_eq @ nat @ ( code_nat_of_natural @ X4 ) @ ( code_nat_of_natural @ Xa3 ) ) ) ) ).
% less_eq_natural.rep_eq
thf(fact_7175_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa2: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X @ Xa2 @ Xb )
= Y )
=> ( ( ( X
= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X
!= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa2 @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa2 ) ) ) ) ) ) ).
% iterate.elims
thf(fact_7176_iterate_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( iterate @ B @ A )
= ( ^ [K3: code_natural,F4: B > A > ( product_prod @ B @ A ),X4: B] :
( if @ ( A > ( product_prod @ B @ A ) )
@ ( K3
= ( zero_zero @ code_natural ) )
@ ( product_Pair @ B @ A @ X4 )
@ ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( F4 @ X4 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ K3 @ ( one_one @ code_natural ) ) @ F4 ) ) ) ) ) ).
% iterate.simps
thf(fact_7177_next_Osimps,axiom,
! [V3: code_natural,W2: code_natural] :
( ( next @ ( product_Pair @ code_natural @ code_natural @ V3 @ W2 ) )
= ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V3 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V3 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( one_one @ code_natural ) ) ) @ ( one_one @ code_natural ) ) @ ( product_Pair @ code_natural @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V3 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V3 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% next.simps
thf(fact_7178_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa2: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X @ Xa2 @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa2 @ Xb ) ) )
=> ~ ( ( ( ( X
= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X
!= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa2 @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa2 @ Xb ) ) ) ) ) ) ).
% iterate.pelims
thf(fact_7179_Random_Orange__def,axiom,
( range
= ( ^ [K3: code_natural] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) )
@ ( iterate @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( log @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K3 )
@ ^ [L2: code_natural] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ next
@ ^ [V4: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ V4 @ ( times_times @ code_natural @ L2 @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ code_natural ) )
@ ^ [V4: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( modulo_modulo @ code_natural @ V4 @ K3 ) ) ) ) ) ).
% Random.range_def
thf(fact_7180_Suc_Orep__eq,axiom,
! [X: code_natural] :
( ( code_nat_of_natural @ ( code_Suc @ X ) )
= ( suc @ ( code_nat_of_natural @ X ) ) ) ).
% Suc.rep_eq
thf(fact_7181_select__def,axiom,
! [A: $tType] :
( ( select @ A )
= ( ^ [Xs: list @ A] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) @ ( range @ ( code_natural_of_nat @ ( size_size @ ( list @ A ) @ Xs ) ) )
@ ^ [K3: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( nth @ A @ Xs @ ( code_nat_of_natural @ K3 ) ) ) ) ) ) ).
% select_def
thf(fact_7182_trancl__def,axiom,
! [A: $tType] :
( ( transitive_trancl @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_tranclp @ A
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R ) ) ) ) ) ) ).
% trancl_def
thf(fact_7183_Suc_Oabs__eq,axiom,
! [X: nat] :
( ( code_Suc @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( suc @ X ) ) ) ).
% Suc.abs_eq
thf(fact_7184_less__eq__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] :
( ( ord_less_eq @ code_natural @ ( code_natural_of_nat @ Xa2 ) @ ( code_natural_of_nat @ X ) )
= ( ord_less_eq @ nat @ Xa2 @ X ) ) ).
% less_eq_natural.abs_eq
thf(fact_7185_times__natural_Oabs__eq,axiom,
! [Xa2: nat,X: nat] :
( ( times_times @ code_natural @ ( code_natural_of_nat @ Xa2 ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( times_times @ nat @ Xa2 @ X ) ) ) ).
% times_natural.abs_eq
thf(fact_7186_less__nat__rel,axiom,
( ( ord_less @ nat )
= ( transitive_tranclp @ nat
@ ^ [M6: nat,N5: nat] :
( N5
= ( suc @ M6 ) ) ) ) ).
% less_nat_rel
thf(fact_7187_zero__natural__def,axiom,
( ( zero_zero @ code_natural )
= ( code_natural_of_nat @ ( zero_zero @ nat ) ) ) ).
% zero_natural_def
thf(fact_7188_tranclp__induct2,axiom,
! [A: $tType,B: $tType,R3: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P2: A > B > $o] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ! [A5: A,B4: B] :
( ( R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P2 @ A5 @ B4 ) )
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( ( R3 @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( P2 @ A5 @ B4 )
=> ( P2 @ Aa2 @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% tranclp_induct2
thf(fact_7189_tranclp__trancl__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_tranclp @ A
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) )
= ( ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% tranclp_trancl_eq
thf(fact_7190_tranclp__power,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [P4: A > A > $o,X4: A,Y4: A] :
? [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( compow @ ( A > A > $o ) @ N5 @ P4 @ X4 @ Y4 ) ) ) ) ).
% tranclp_power
thf(fact_7191_Nitpick_Otranclp__unfold,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [R: A > A > $o,A7: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ ( transitive_trancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R ) ) ) ) ) ) ).
% Nitpick.tranclp_unfold
thf(fact_7192_pick__same,axiom,
! [A: $tType,L: nat,Xs2: list @ A] :
( ( ord_less @ nat @ L @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( pick @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs2 ) @ ( code_natural_of_nat @ L ) )
= ( nth @ A @ Xs2 @ L ) ) ) ).
% pick_same
thf(fact_7193_Code__Numeral_OSuc__def,axiom,
( code_Suc
= ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).
% Code_Numeral.Suc_def
thf(fact_7194_times__natural__def,axiom,
( ( times_times @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( times_times @ nat ) ) ) ).
% times_natural_def
thf(fact_7195_map__option__o__case__sum,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F3: D > C,G3: A > ( option @ D ),H: B > ( option @ D )] :
( ( comp @ ( option @ D ) @ ( option @ C ) @ ( sum_sum @ A @ B ) @ ( map_option @ D @ C @ F3 ) @ ( sum_case_sum @ A @ ( option @ D ) @ B @ G3 @ H ) )
= ( sum_case_sum @ A @ ( option @ C ) @ B @ ( comp @ ( option @ D ) @ ( option @ C ) @ A @ ( map_option @ D @ C @ F3 ) @ G3 ) @ ( comp @ ( option @ D ) @ ( option @ C ) @ B @ ( map_option @ D @ C @ F3 ) @ H ) ) ) ).
% map_option_o_case_sum
thf(fact_7196_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,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( F3 @ X4 ) @ Y4 ) )
@ ( none @ A )
@ A6 ) ) ) ) ).
% semilattice_set.eq_fold'
thf(fact_7197_linear__injective__on__subspace__0,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F3: A > B,S2: set @ A] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( ( real_Vector_subspace @ A @ S2 )
=> ( ( inj_on @ A @ B @ F3 @ S2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( ( F3 @ X4 )
= ( zero_zero @ B ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ) ) ) ).
% linear_injective_on_subspace_0
thf(fact_7198_subspace__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( real_Vector_subspace @ A @ S3 )
=> ( member @ A @ ( zero_zero @ A ) @ S3 ) ) ) ).
% subspace_0
thf(fact_7199_linear__subspace__kernel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F3: A > B] :
( ( real_Vector_linear @ A @ B @ F3 )
=> ( real_Vector_subspace @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
= ( zero_zero @ B ) ) ) ) ) ) ).
% linear_subspace_kernel
thf(fact_7200_subspace__single__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( real_Vector_subspace @ A @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% subspace_single_0
thf(fact_7201_subspaceI,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ S3 )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( member @ A @ Y3 @ S3 )
=> ( member @ A @ ( plus_plus @ A @ X3 @ Y3 ) @ S3 ) ) )
=> ( ! [C2: real,X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( member @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ X3 ) @ S3 ) )
=> ( real_Vector_subspace @ A @ S3 ) ) ) ) ) ).
% subspaceI
thf(fact_7202_subspace__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_subspace @ A )
= ( ^ [S7: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ S7 )
& ! [X4: A] :
( ( member @ A @ X4 @ S7 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ S7 )
=> ( member @ A @ ( plus_plus @ A @ X4 @ Y4 ) @ S7 ) ) )
& ! [C4: real,X4: A] :
( ( member @ A @ X4 @ S7 )
=> ( member @ A @ ( real_V8093663219630862766scaleR @ A @ C4 @ X4 ) @ S7 ) ) ) ) ) ) ).
% subspace_def
thf(fact_7203_old_Orec__bool__def,axiom,
! [T: $tType] :
( ( product_rec_bool @ T )
= ( ^ [F12: T,F23: T,X4: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F23 @ X4 ) ) ) ) ).
% old.rec_bool_def
thf(fact_7204_ordering__top_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( ordering_top_axioms @ A @ Less_eq2 @ Top ) ) ).
% ordering_top.axioms(2)
thf(fact_7205_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $true )
= F1 ) ).
% old.bool.simps(5)
thf(fact_7206_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $false )
= F22 ) ).
% old.bool.simps(6)
thf(fact_7207_ordering__top__axioms__def,axiom,
! [A: $tType] :
( ( ordering_top_axioms @ A )
= ( ^ [Less_eq: A > A > $o,Top2: A] :
! [A7: A] : ( Less_eq @ A7 @ Top2 ) ) ) ).
% ordering_top_axioms_def
thf(fact_7208_ordering__top__axioms_Ointro,axiom,
! [A: $tType,Less_eq2: A > A > $o,Top: A] :
( ! [A5: A] : ( Less_eq2 @ A5 @ Top )
=> ( ordering_top_axioms @ A @ Less_eq2 @ Top ) ) ).
% ordering_top_axioms.intro
thf(fact_7209_ordering__top__def,axiom,
! [A: $tType] :
( ( ordering_top @ A )
= ( ^ [Less_eq: A > A > $o,Less2: A > A > $o,Top2: A] :
( ( ordering @ A @ Less_eq @ Less2 )
& ( ordering_top_axioms @ A @ Less_eq @ Top2 ) ) ) ) ).
% ordering_top_def
thf(fact_7210_ordering__top_Ointro,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A] :
( ( ordering @ A @ Less_eq2 @ Less )
=> ( ( ordering_top_axioms @ A @ Less_eq2 @ Top )
=> ( ordering_top @ A @ Less_eq2 @ Less @ Top ) ) ) ).
% ordering_top.intro
thf(fact_7211_ordering_Oeq__iff,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq2 @ Less )
=> ( ( A3 = B2 )
= ( ( Less_eq2 @ A3 @ B2 )
& ( Less_eq2 @ B2 @ A3 ) ) ) ) ).
% ordering.eq_iff
thf(fact_7212_ordering_Oantisym,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq2 @ Less )
=> ( ( Less_eq2 @ A3 @ B2 )
=> ( ( Less_eq2 @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% ordering.antisym
thf(fact_7213_ordering_Oorder__iff__strict,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq2 @ Less )
=> ( ( Less_eq2 @ A3 @ B2 )
= ( ( Less @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% ordering.order_iff_strict
thf(fact_7214_ordering_Ostrict__iff__order,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq2 @ Less )
=> ( ( Less @ A3 @ B2 )
= ( ( Less_eq2 @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% ordering.strict_iff_order
thf(fact_7215_ordering_Ostrict__implies__not__eq,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq2 @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% ordering.strict_implies_not_eq
thf(fact_7216_ordering_Onot__eq__order__implies__strict,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq2 @ Less )
=> ( ( A3 != B2 )
=> ( ( Less_eq2 @ A3 @ B2 )
=> ( Less @ A3 @ B2 ) ) ) ) ).
% ordering.not_eq_order_implies_strict
thf(fact_7217_ordering__strictI,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o] :
( ! [A5: A,B4: A] :
( ( Less_eq2 @ A5 @ B4 )
= ( ( Less @ A5 @ B4 )
| ( A5 = B4 ) ) )
=> ( ! [A5: A,B4: A] :
( ( Less @ A5 @ B4 )
=> ~ ( Less @ B4 @ A5 ) )
=> ( ! [A5: A] :
~ ( Less @ A5 @ A5 )
=> ( ! [A5: A,B4: A,C2: A] :
( ( Less @ A5 @ B4 )
=> ( ( Less @ B4 @ C2 )
=> ( Less @ A5 @ C2 ) ) )
=> ( ordering @ A @ Less_eq2 @ Less ) ) ) ) ) ).
% ordering_strictI
thf(fact_7218_ordering__dualI,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o] :
( ( ordering @ A
@ ^ [A7: A,B5: A] : ( Less_eq2 @ B5 @ A7 )
@ ^ [A7: A,B5: A] : ( Less @ B5 @ A7 ) )
=> ( ordering @ A @ Less_eq2 @ Less ) ) ).
% ordering_dualI
thf(fact_7219_ordering__top_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,Top: A] :
( ( ordering_top @ A @ Less_eq2 @ Less @ Top )
=> ( ordering @ A @ Less_eq2 @ Less ) ) ).
% ordering_top.axioms(1)
thf(fact_7220_order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.ordering_axioms
thf(fact_7221_dual__order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ) ).
% dual_order.ordering_axioms
thf(fact_7222_eventually__filtercomap__at__bot__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P2: B > $o,F3: B > A] :
( ( eventually @ B @ P2 @ ( filtercomap @ B @ A @ F3 @ ( at_bot @ A ) ) )
= ( ? [N6: A] :
! [X4: B] :
( ( ord_less_eq @ A @ ( F3 @ X4 ) @ N6 )
=> ( P2 @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_linorder
thf(fact_7223_list__all__length,axiom,
! [A: $tType] :
( ( list_all @ A )
= ( ^ [P4: A > $o,Xs: list @ A] :
! [N5: nat] :
( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P4 @ ( nth @ A @ Xs @ N5 ) ) ) ) ) ).
% list_all_length
thf(fact_7224_eventually__filtercomap__at__top__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P2: B > $o,F3: B > A] :
( ( eventually @ B @ P2 @ ( filtercomap @ B @ A @ F3 @ ( at_top @ A ) ) )
= ( ? [N6: A] :
! [X4: B] :
( ( ord_less_eq @ A @ N6 @ ( F3 @ X4 ) )
=> ( P2 @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_top_linorder
thf(fact_7225_prod__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups1828464146339083142d_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod_list.comm_monoid_list_axioms
thf(fact_7226_Pair__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A6: A > B > $o,B6: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( C > ( product_prod @ A @ C ) ) @ ( D > ( product_prod @ B @ D ) ) @ A6 @ ( bNF_rel_fun @ C @ D @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ B6 @ ( basic_rel_prod @ A @ B @ C @ D @ A6 @ B6 ) ) @ ( product_Pair @ A @ C ) @ ( product_Pair @ B @ D ) ) ).
% Pair_transfer
thf(fact_7227_sum__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups1828464146339083142d_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum_list.comm_monoid_list_axioms
thf(fact_7228_VEBT_Orec__transfer,axiom,
! [A: $tType,B: $tType,S3: A > B > $o] :
( bNF_rel_fun @ ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A ) @ ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ ( product_prod @ vEBT_VEBT @ B ) ) > vEBT_VEBT > B > B ) @ ( ( $o > $o > A ) > vEBT_VEBT > A ) @ ( ( $o > $o > B ) > vEBT_VEBT > B )
@ ( bNF_rel_fun @ ( option @ ( product_prod @ nat @ nat ) ) @ ( option @ ( product_prod @ nat @ nat ) ) @ ( nat > ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A ) @ ( nat > ( list @ ( product_prod @ vEBT_VEBT @ B ) ) > vEBT_VEBT > B > B )
@ ^ [Y5: option @ ( product_prod @ nat @ nat ),Z2: option @ ( product_prod @ nat @ nat )] : Y5 = Z2
@ ( bNF_rel_fun @ nat @ nat @ ( ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A ) @ ( ( list @ ( product_prod @ vEBT_VEBT @ B ) ) > vEBT_VEBT > B > B )
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ( bNF_rel_fun @ ( list @ ( product_prod @ vEBT_VEBT @ A ) ) @ ( list @ ( product_prod @ vEBT_VEBT @ B ) ) @ ( vEBT_VEBT > A > A ) @ ( vEBT_VEBT > B > B )
@ ( list_all2 @ ( product_prod @ vEBT_VEBT @ A ) @ ( product_prod @ vEBT_VEBT @ B )
@ ( basic_rel_prod @ vEBT_VEBT @ vEBT_VEBT @ A @ B
@ ^ [Y5: vEBT_VEBT,Z2: vEBT_VEBT] : Y5 = Z2
@ S3 ) )
@ ( bNF_rel_fun @ vEBT_VEBT @ vEBT_VEBT @ ( A > A ) @ ( B > B )
@ ^ [Y5: vEBT_VEBT,Z2: vEBT_VEBT] : Y5 = Z2
@ ( bNF_rel_fun @ A @ B @ A @ B @ S3 @ S3 ) ) ) ) )
@ ( bNF_rel_fun @ ( $o > $o > A ) @ ( $o > $o > B ) @ ( vEBT_VEBT > A ) @ ( vEBT_VEBT > B )
@ ( bNF_rel_fun @ $o @ $o @ ( $o > A ) @ ( $o > B )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ S3 ) )
@ ( bNF_rel_fun @ vEBT_VEBT @ vEBT_VEBT @ A @ B
@ ^ [Y5: vEBT_VEBT,Z2: vEBT_VEBT] : Y5 = Z2
@ S3 ) )
@ ( vEBT_rec_VEBT @ A )
@ ( vEBT_rec_VEBT @ B ) ) ).
% VEBT.rec_transfer
thf(fact_7229_rel__prod__inject,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: A > B > $o,R23: C > D > $o,A3: A,B2: C,C3: B,D3: D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A3 @ B2 ) @ ( product_Pair @ B @ D @ C3 @ D3 ) )
= ( ( R12 @ A3 @ C3 )
& ( R23 @ B2 @ D3 ) ) ) ).
% rel_prod_inject
thf(fact_7230_rel__prod_Ocases,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: A > B > $o,R23: C > D > $o,A12: product_prod @ A @ C,A23: product_prod @ B @ D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ A12 @ A23 )
=> ~ ! [A5: A,B4: B,C2: C] :
( ( A12
= ( product_Pair @ A @ C @ A5 @ C2 ) )
=> ! [D2: D] :
( ( A23
= ( product_Pair @ B @ D @ B4 @ D2 ) )
=> ( ( R12 @ A5 @ B4 )
=> ~ ( R23 @ C2 @ D2 ) ) ) ) ) ).
% rel_prod.cases
thf(fact_7231_rel__prod_Osimps,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( basic_rel_prod @ A @ B @ C @ D )
= ( ^ [R13: A > B > $o,R24: C > D > $o,A13: product_prod @ A @ C,A24: product_prod @ B @ D] :
? [A7: A,B5: B,C4: C,D4: D] :
( ( A13
= ( product_Pair @ A @ C @ A7 @ C4 ) )
& ( A24
= ( product_Pair @ B @ D @ B5 @ D4 ) )
& ( R13 @ A7 @ B5 )
& ( R24 @ C4 @ D4 ) ) ) ) ).
% rel_prod.simps
thf(fact_7232_rel__prod_Ointros,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R12: A > B > $o,A3: A,B2: B,R23: C > D > $o,C3: C,D3: D] :
( ( R12 @ A3 @ B2 )
=> ( ( R23 @ C3 @ D3 )
=> ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A3 @ C3 ) @ ( product_Pair @ B @ D @ B2 @ D3 ) ) ) ) ).
% rel_prod.intros
thf(fact_7233_VEBT_Ocase__transfer,axiom,
! [A: $tType,B: $tType,S3: A > B > $o] :
( bNF_rel_fun @ ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A ) @ ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > B ) @ ( ( $o > $o > A ) > vEBT_VEBT > A ) @ ( ( $o > $o > B ) > vEBT_VEBT > B )
@ ( bNF_rel_fun @ ( option @ ( product_prod @ nat @ nat ) ) @ ( option @ ( product_prod @ nat @ nat ) ) @ ( nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A ) @ ( nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > B )
@ ^ [Y5: option @ ( product_prod @ nat @ nat ),Z2: option @ ( product_prod @ nat @ nat )] : Y5 = Z2
@ ( bNF_rel_fun @ nat @ nat @ ( ( list @ vEBT_VEBT ) > vEBT_VEBT > A ) @ ( ( list @ vEBT_VEBT ) > vEBT_VEBT > B )
@ ^ [Y5: nat,Z2: nat] : Y5 = Z2
@ ( bNF_rel_fun @ ( list @ vEBT_VEBT ) @ ( list @ vEBT_VEBT ) @ ( vEBT_VEBT > A ) @ ( vEBT_VEBT > B )
@ ^ [Y5: list @ vEBT_VEBT,Z2: list @ vEBT_VEBT] : Y5 = Z2
@ ( bNF_rel_fun @ vEBT_VEBT @ vEBT_VEBT @ A @ B
@ ^ [Y5: vEBT_VEBT,Z2: vEBT_VEBT] : Y5 = Z2
@ S3 ) ) ) )
@ ( bNF_rel_fun @ ( $o > $o > A ) @ ( $o > $o > B ) @ ( vEBT_VEBT > A ) @ ( vEBT_VEBT > B )
@ ( bNF_rel_fun @ $o @ $o @ ( $o > A ) @ ( $o > B )
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y5: $o,Z2: $o] : Y5 = Z2
@ S3 ) )
@ ( bNF_rel_fun @ vEBT_VEBT @ vEBT_VEBT @ A @ B
@ ^ [Y5: vEBT_VEBT,Z2: vEBT_VEBT] : Y5 = Z2
@ S3 ) )
@ ( vEBT_case_VEBT @ A )
@ ( vEBT_case_VEBT @ B ) ) ).
% VEBT.case_transfer
thf(fact_7234_is__arg__max__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ( ( lattic501386751176901750rg_max @ A @ B )
= ( ^ [F4: A > B,P4: A > $o,X4: A] :
( ( P4 @ X4 )
& ! [Y4: A] :
( ( P4 @ Y4 )
=> ( ord_less_eq @ B @ ( F4 @ Y4 ) @ ( F4 @ X4 ) ) ) ) ) ) ) ).
% is_arg_max_linorder
thf(fact_7235_VEBT_Osimps_I6_J,axiom,
! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A,F22: $o > $o > A,X21: $o,X22: $o] :
( ( vEBT_case_VEBT @ A @ F1 @ F22 @ ( vEBT_Leaf @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% VEBT.simps(6)
thf(fact_7236_VEBT_Osimps_I5_J,axiom,
! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A,F22: $o > $o > A,X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( vEBT_case_VEBT @ A @ F1 @ F22 @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( F1 @ X11 @ X12 @ X13 @ X14 ) ) ).
% VEBT.simps(5)
thf(fact_7237_VEBT_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A,F22: $o > $o > A,VEBT2: vEBT_VEBT] :
( ( H @ ( vEBT_case_VEBT @ A @ F1 @ F22 @ VEBT2 ) )
= ( vEBT_case_VEBT @ B
@ ^ [X16: option @ ( product_prod @ nat @ nat ),X24: nat,X34: list @ vEBT_VEBT,X43: vEBT_VEBT] : ( H @ ( F1 @ X16 @ X24 @ X34 @ X43 ) )
@ ^ [X16: $o,X24: $o] : ( H @ ( F22 @ X16 @ X24 ) )
@ VEBT2 ) ) ).
% VEBT.case_distrib
thf(fact_7238_group_Oaxioms_I2_J,axiom,
! [A: $tType,F3: A > A > A,Z3: A,Inverse: A > A] :
( ( group @ A @ F3 @ Z3 @ Inverse )
=> ( group_axioms @ A @ F3 @ Z3 @ Inverse ) ) ).
% group.axioms(2)
thf(fact_7239_is__none__bind,axiom,
! [A: $tType,B: $tType,F3: option @ B,G3: B > ( option @ A )] :
( ( is_none @ A @ ( bind @ B @ A @ F3 @ G3 ) )
= ( ( is_none @ B @ F3 )
| ( is_none @ A @ ( G3 @ ( the2 @ B @ F3 ) ) ) ) ) ).
% is_none_bind
thf(fact_7240_is__none__code_I2_J,axiom,
! [B: $tType,X: B] :
~ ( is_none @ B @ ( some @ B @ X ) ) ).
% is_none_code(2)
thf(fact_7241_is__none__code_I1_J,axiom,
! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% is_none_code(1)
thf(fact_7242_is__none__map__option,axiom,
! [A: $tType,B: $tType,F3: B > A,X: option @ B] :
( ( is_none @ A @ ( map_option @ B @ A @ F3 @ X ) )
= ( is_none @ B @ X ) ) ).
% is_none_map_option
thf(fact_7243_group__axioms__def,axiom,
! [A: $tType] :
( ( group_axioms @ A )
= ( ^ [F4: A > A > A,Z4: A,Inverse2: A > A] :
( ! [A7: A] :
( ( F4 @ Z4 @ A7 )
= A7 )
& ! [A7: A] :
( ( F4 @ ( Inverse2 @ A7 ) @ A7 )
= Z4 ) ) ) ) ).
% group_axioms_def
thf(fact_7244_group__axioms_Ointro,axiom,
! [A: $tType,F3: A > A > A,Z3: A,Inverse: A > A] :
( ! [A5: A] :
( ( F3 @ Z3 @ A5 )
= A5 )
=> ( ! [A5: A] :
( ( F3 @ ( Inverse @ A5 ) @ A5 )
= Z3 )
=> ( group_axioms @ A @ F3 @ Z3 @ Inverse ) ) ) ).
% group_axioms.intro
thf(fact_7245_Option_Ois__none__def,axiom,
! [A: $tType] :
( ( is_none @ A )
= ( ^ [X4: option @ A] :
( X4
= ( none @ A ) ) ) ) ).
% Option.is_none_def
thf(fact_7246_is__none__simps_I1_J,axiom,
! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% is_none_simps(1)
thf(fact_7247_is__none__simps_I2_J,axiom,
! [B: $tType,X: B] :
~ ( is_none @ B @ ( some @ B @ X ) ) ).
% is_none_simps(2)
thf(fact_7248_the__map__option,axiom,
! [B: $tType,A: $tType,X: option @ A,F3: A > B] :
( ~ ( is_none @ A @ X )
=> ( ( the2 @ B @ ( map_option @ A @ B @ F3 @ X ) )
= ( F3 @ ( the2 @ A @ X ) ) ) ) ).
% the_map_option
thf(fact_7249_rel__option__unfold,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,X4: option @ A,Y4: option @ B] :
( ( ( is_none @ A @ X4 )
= ( is_none @ B @ Y4 ) )
& ( ~ ( is_none @ A @ X4 )
=> ( ~ ( is_none @ B @ Y4 )
=> ( R6 @ ( the2 @ A @ X4 ) @ ( the2 @ B @ Y4 ) ) ) ) ) ) ) ).
% rel_option_unfold
thf(fact_7250_rel__optionI,axiom,
! [A: $tType,B: $tType,X: option @ A,Y: option @ B,P2: A > B > $o] :
( ( ( is_none @ A @ X )
= ( is_none @ B @ Y ) )
=> ( ( ~ ( is_none @ A @ X )
=> ( ~ ( is_none @ B @ Y )
=> ( P2 @ ( the2 @ A @ X ) @ ( the2 @ B @ Y ) ) ) )
=> ( rel_option @ A @ B @ P2 @ X @ Y ) ) ) ).
% rel_optionI
thf(fact_7251_group_Ointro,axiom,
! [A: $tType,F3: A > A > A,Z3: A,Inverse: A > A] :
( ( semigroup @ A @ F3 )
=> ( ( group_axioms @ A @ F3 @ Z3 @ Inverse )
=> ( group @ A @ F3 @ Z3 @ Inverse ) ) ) ).
% group.intro
thf(fact_7252_group__def,axiom,
! [A: $tType] :
( ( group @ A )
= ( ^ [F4: A > A > A,Z4: A,Inverse2: A > A] :
( ( semigroup @ A @ F4 )
& ( group_axioms @ A @ F4 @ Z4 @ Inverse2 ) ) ) ) ).
% group_def
thf(fact_7253_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.semigroup_axioms
thf(fact_7254_semigroup_Oassoc,axiom,
! [A: $tType,F3: A > A > A,A3: A,B2: A,C3: A] :
( ( semigroup @ A @ F3 )
=> ( ( F3 @ ( F3 @ A3 @ B2 ) @ C3 )
= ( F3 @ A3 @ ( F3 @ B2 @ C3 ) ) ) ) ).
% semigroup.assoc
thf(fact_7255_semigroup_Ointro,axiom,
! [A: $tType,F3: A > A > A] :
( ! [A5: A,B4: A,C2: A] :
( ( F3 @ ( F3 @ A5 @ B4 ) @ C2 )
= ( F3 @ A5 @ ( F3 @ B4 @ C2 ) ) )
=> ( semigroup @ A @ F3 ) ) ).
% semigroup.intro
thf(fact_7256_semigroup__def,axiom,
! [A: $tType] :
( ( semigroup @ A )
= ( ^ [F4: A > A > A] :
! [A7: A,B5: A,C4: A] :
( ( F4 @ ( F4 @ A7 @ B5 ) @ C4 )
= ( F4 @ A7 @ ( F4 @ B5 @ C4 ) ) ) ) ) ).
% semigroup_def
thf(fact_7257_group_Oaxioms_I1_J,axiom,
! [A: $tType,F3: A > A > A,Z3: A,Inverse: A > A] :
( ( group @ A @ F3 @ Z3 @ Inverse )
=> ( semigroup @ A @ F3 ) ) ).
% group.axioms(1)
thf(fact_7258_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ( semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.semigroup_axioms
thf(fact_7259_ordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o] :
( ( ordering @ A @ Less_eq2 @ Less )
=> ( ordering_axioms @ A @ Less_eq2 @ Less ) ) ).
% ordering.axioms(2)
thf(fact_7260_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F3 ) )
= F3 ) ).
% case_prod_curry
thf(fact_7261_curryI,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
( ( F3 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( product_curry @ A @ B @ $o @ F3 @ A3 @ B2 ) ) ).
% curryI
thf(fact_7262_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F4: ( product_prod @ B @ C ) > A,A7: B,B5: C] : ( F4 @ ( product_Pair @ B @ C @ A7 @ B5 ) ) ) ) ).
% curry_conv
thf(fact_7263_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F3 ) )
= F3 ) ).
% curry_case_prod
thf(fact_7264_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C4: ( product_prod @ A @ B ) > C,X4: A,Y4: B] : ( C4 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) ) ) ).
% curry_def
thf(fact_7265_curryE,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F3 @ A3 @ B2 )
=> ( F3 @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% curryE
thf(fact_7266_curryD,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F3 @ A3 @ B2 )
=> ( F3 @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% curryD
thf(fact_7267_curry__K,axiom,
! [B: $tType,C: $tType,A: $tType,C3: C] :
( ( product_curry @ A @ B @ C
@ ^ [X4: product_prod @ A @ B] : C3 )
= ( ^ [X4: A,Y4: B] : C3 ) ) ).
% curry_K
thf(fact_7268_ordering__axioms__def,axiom,
! [A: $tType] :
( ( ordering_axioms @ A )
= ( ^ [Less_eq: A > A > $o,Less2: A > A > $o] :
( ! [A7: A,B5: A] :
( ( Less2 @ A7 @ B5 )
= ( ( Less_eq @ A7 @ B5 )
& ( A7 != B5 ) ) )
& ! [A7: A,B5: A] :
( ( Less_eq @ A7 @ B5 )
=> ( ( Less_eq @ B5 @ A7 )
=> ( A7 = B5 ) ) ) ) ) ) ).
% ordering_axioms_def
thf(fact_7269_ordering__axioms_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq2: A > A > $o] :
( ! [A5: A,B4: A] :
( ( Less @ A5 @ B4 )
= ( ( Less_eq2 @ A5 @ B4 )
& ( A5 != B4 ) ) )
=> ( ! [A5: A,B4: A] :
( ( Less_eq2 @ A5 @ B4 )
=> ( ( Less_eq2 @ B4 @ A5 )
=> ( A5 = B4 ) ) )
=> ( ordering_axioms @ A @ Less_eq2 @ Less ) ) ) ).
% ordering_axioms.intro
thf(fact_7270_ordering_Ointro,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o] :
( ( partial_preordering @ A @ Less_eq2 )
=> ( ( ordering_axioms @ A @ Less_eq2 @ Less )
=> ( ordering @ A @ Less_eq2 @ Less ) ) ) ).
% ordering.intro
thf(fact_7271_ordering__def,axiom,
! [A: $tType] :
( ( ordering @ A )
= ( ^ [Less_eq: A > A > $o,Less2: A > A > $o] :
( ( partial_preordering @ A @ Less_eq )
& ( ordering_axioms @ A @ Less_eq @ Less2 ) ) ) ) ).
% ordering_def
thf(fact_7272_partial__preordering_Orefl,axiom,
! [A: $tType,Less_eq2: A > A > $o,A3: A] :
( ( partial_preordering @ A @ Less_eq2 )
=> ( Less_eq2 @ A3 @ A3 ) ) ).
% partial_preordering.refl
thf(fact_7273_partial__preordering_Ointro,axiom,
! [A: $tType,Less_eq2: A > A > $o] :
( ! [A5: A] : ( Less_eq2 @ A5 @ A5 )
=> ( ! [A5: A,B4: A,C2: A] :
( ( Less_eq2 @ A5 @ B4 )
=> ( ( Less_eq2 @ B4 @ C2 )
=> ( Less_eq2 @ A5 @ C2 ) ) )
=> ( partial_preordering @ A @ Less_eq2 ) ) ) ).
% partial_preordering.intro
thf(fact_7274_partial__preordering_Otrans,axiom,
! [A: $tType,Less_eq2: A > A > $o,A3: A,B2: A,C3: A] :
( ( partial_preordering @ A @ Less_eq2 )
=> ( ( Less_eq2 @ A3 @ B2 )
=> ( ( Less_eq2 @ B2 @ C3 )
=> ( Less_eq2 @ A3 @ C3 ) ) ) ) ).
% partial_preordering.trans
thf(fact_7275_partial__preordering__def,axiom,
! [A: $tType] :
( ( partial_preordering @ A )
= ( ^ [Less_eq: A > A > $o] :
( ! [A7: A] : ( Less_eq @ A7 @ A7 )
& ! [A7: A,B5: A,C4: A] :
( ( Less_eq @ A7 @ B5 )
=> ( ( Less_eq @ B5 @ C4 )
=> ( Less_eq @ A7 @ C4 ) ) ) ) ) ) ).
% partial_preordering_def
thf(fact_7276_dual__order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( partial_preordering @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_7277_order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( partial_preordering @ A @ ( ord_less_eq @ A ) ) ) ).
% order.partial_preordering_axioms
thf(fact_7278_ordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o] :
( ( ordering @ A @ Less_eq2 @ Less )
=> ( partial_preordering @ A @ Less_eq2 ) ) ).
% ordering.axioms(1)
thf(fact_7279_wfP__wf__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( wfP @ A
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) )
= ( wf @ A @ R3 ) ) ).
% wfP_wf_eq
thf(fact_7280_prod__list_Omonoid__list__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( groups_monoid_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod_list.monoid_list_axioms
thf(fact_7281_sum__list_Omonoid__list__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( groups_monoid_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum_list.monoid_list_axioms
thf(fact_7282_typerep_Osize__neq,axiom,
! [X: typerep] :
( ( size_size @ typerep @ X )
!= ( zero_zero @ nat ) ) ).
% typerep.size_neq
thf(fact_7283_tuple__isomorphism_Osize__neq,axiom,
! [A: $tType,B: $tType,C: $tType,X: tuple_isomorphism @ A @ B @ C] :
( ( size_size @ ( tuple_isomorphism @ A @ B @ C ) @ X )
!= ( zero_zero @ nat ) ) ).
% tuple_isomorphism.size_neq
thf(fact_7284_tuple__isomorphism_Osize_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,X1: A > ( product_prod @ B @ C ),X2: ( product_prod @ B @ C ) > A] :
( ( size_size @ ( tuple_isomorphism @ A @ B @ C ) @ ( tuple_1188178415141063261rphism @ A @ B @ C @ X1 @ X2 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% tuple_isomorphism.size(2)
thf(fact_7285_typerep_Osize_I2_J,axiom,
! [X1: literal,X2: list @ typerep] :
( ( size_size @ typerep @ ( typerep2 @ X1 @ X2 ) )
= ( plus_plus @ nat @ ( size_list @ typerep @ ( size_size @ typerep ) @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% typerep.size(2)
thf(fact_7286_typerep_Osize__gen,axiom,
! [X1: literal,X2: list @ typerep] :
( ( size_typerep @ ( typerep2 @ X1 @ X2 ) )
= ( plus_plus @ nat @ ( size_list @ typerep @ size_typerep @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% typerep.size_gen
thf(fact_7287_tuple__isomorphism_Osize__gen,axiom,
! [B: $tType,C: $tType,A: $tType,Xb: A > nat,Xa2: B > nat,X: C > nat,X1: A > ( product_prod @ B @ C ),X2: ( product_prod @ B @ C ) > A] :
( ( tuple_9181185373184732606rphism @ A @ B @ C @ Xb @ Xa2 @ X @ ( tuple_1188178415141063261rphism @ A @ B @ C @ X1 @ X2 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% tuple_isomorphism.size_gen
thf(fact_7288_max__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.semilattice_neutr_axioms
thf(fact_7289_curr__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( bNF_Wellorder_curr @ A @ B @ C )
= ( ^ [A8: set @ A,F4: ( product_prod @ A @ B ) > C,A7: A] :
( if @ ( B > C ) @ ( member @ A @ A7 @ A8 )
@ ^ [B5: B] : ( F4 @ ( product_Pair @ A @ B @ A7 @ B5 ) )
@ ( undefined @ ( B > C ) ) ) ) ) ).
% curr_def
thf(fact_7290_gcd__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.semilattice_neutr_axioms
thf(fact_7291_or_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( semilattice_neutr @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.semilattice_neutr_axioms
thf(fact_7292_dual__order_Opreordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( preordering @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ) ).
% dual_order.preordering_axioms
thf(fact_7293_closed__Collect__le,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,G3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G3 )
=> ( topolo7761053866217962861closed @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) ) ) ) ) ) ).
% closed_Collect_le
thf(fact_7294_preordering_Oasym,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( preordering @ A @ Less_eq2 @ Less )
=> ( ( Less @ A3 @ B2 )
=> ~ ( Less @ B2 @ A3 ) ) ) ).
% preordering.asym
thf(fact_7295_preordering_Oirrefl,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,A3: A] :
( ( preordering @ A @ Less_eq2 @ Less )
=> ~ ( Less @ A3 @ A3 ) ) ).
% preordering.irrefl
thf(fact_7296_preordering_Ostrict__trans,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,A3: A,B2: A,C3: A] :
( ( preordering @ A @ Less_eq2 @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( ( Less @ B2 @ C3 )
=> ( Less @ A3 @ C3 ) ) ) ) ).
% preordering.strict_trans
thf(fact_7297_preordering_Ostrict__trans1,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,A3: A,B2: A,C3: A] :
( ( preordering @ A @ Less_eq2 @ Less )
=> ( ( Less_eq2 @ A3 @ B2 )
=> ( ( Less @ B2 @ C3 )
=> ( Less @ A3 @ C3 ) ) ) ) ).
% preordering.strict_trans1
thf(fact_7298_preordering_Ostrict__trans2,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,A3: A,B2: A,C3: A] :
( ( preordering @ A @ Less_eq2 @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( ( Less_eq2 @ B2 @ C3 )
=> ( Less @ A3 @ C3 ) ) ) ) ).
% preordering.strict_trans2
thf(fact_7299_preordering_Ostrict__iff__not,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( preordering @ A @ Less_eq2 @ Less )
=> ( ( Less @ A3 @ B2 )
= ( ( Less_eq2 @ A3 @ B2 )
& ~ ( Less_eq2 @ B2 @ A3 ) ) ) ) ).
% preordering.strict_iff_not
thf(fact_7300_preordering_Ostrict__implies__order,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( preordering @ A @ Less_eq2 @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( Less_eq2 @ A3 @ B2 ) ) ) ).
% preordering.strict_implies_order
thf(fact_7301_preordering__strictI,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o] :
( ! [A5: A,B4: A] :
( ( Less_eq2 @ A5 @ B4 )
= ( ( Less @ A5 @ B4 )
| ( A5 = B4 ) ) )
=> ( ! [A5: A,B4: A] :
( ( Less @ A5 @ B4 )
=> ~ ( Less @ B4 @ A5 ) )
=> ( ! [A5: A] :
~ ( Less @ A5 @ A5 )
=> ( ! [A5: A,B4: A,C2: A] :
( ( Less @ A5 @ B4 )
=> ( ( Less @ B4 @ C2 )
=> ( Less @ A5 @ C2 ) ) )
=> ( preordering @ A @ Less_eq2 @ Less ) ) ) ) ) ).
% preordering_strictI
thf(fact_7302_preordering__dualI,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o] :
( ( preordering @ A
@ ^ [A7: A,B5: A] : ( Less_eq2 @ B5 @ A7 )
@ ^ [A7: A,B5: A] : ( Less @ B5 @ A7 ) )
=> ( preordering @ A @ Less_eq2 @ Less ) ) ).
% preordering_dualI
thf(fact_7303_order_Opreordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( preordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.preordering_axioms
thf(fact_7304_closed__diagonal,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Y4: product_prod @ A @ A] :
? [X4: A] :
( Y4
= ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ) ).
% closed_diagonal
thf(fact_7305_preordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o] :
( ( preordering @ A @ Less_eq2 @ Less )
=> ( partial_preordering @ A @ Less_eq2 ) ) ).
% preordering.axioms(1)
thf(fact_7306_closed__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y4: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y4 ) )
& ( ord_less_eq @ A @ X4 @ Y4 ) ) ) ) ) ).
% closed_subdiagonal
thf(fact_7307_closed__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y4: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y4 ) )
& ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% closed_superdiagonal
thf(fact_7308_preordering__def,axiom,
! [A: $tType] :
( ( preordering @ A )
= ( ^ [Less_eq: A > A > $o,Less2: A > A > $o] :
( ( partial_preordering @ A @ Less_eq )
& ( preordering_axioms @ A @ Less_eq @ Less2 ) ) ) ) ).
% preordering_def
thf(fact_7309_preordering_Ointro,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o] :
( ( partial_preordering @ A @ Less_eq2 )
=> ( ( preordering_axioms @ A @ Less_eq2 @ Less )
=> ( preordering @ A @ Less_eq2 @ Less ) ) ) ).
% preordering.intro
thf(fact_7310_preordering__axioms__def,axiom,
! [A: $tType] :
( ( preordering_axioms @ A )
= ( ^ [Less_eq: A > A > $o,Less2: A > A > $o] :
! [A7: A,B5: A] :
( ( Less2 @ A7 @ B5 )
= ( ( Less_eq @ A7 @ B5 )
& ~ ( Less_eq @ B5 @ A7 ) ) ) ) ) ).
% preordering_axioms_def
thf(fact_7311_preordering__axioms_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq2: A > A > $o] :
( ! [A5: A,B4: A] :
( ( Less @ A5 @ B4 )
= ( ( Less_eq2 @ A5 @ B4 )
& ~ ( Less_eq2 @ B4 @ A5 ) ) )
=> ( preordering_axioms @ A @ Less_eq2 @ Less ) ) ).
% preordering_axioms.intro
thf(fact_7312_preordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o] :
( ( preordering @ A @ Less_eq2 @ Less )
=> ( preordering_axioms @ A @ Less_eq2 @ Less ) ) ).
% preordering.axioms(2)
thf(fact_7313_real__times__code,axiom,
! [X: rat,Y: rat] :
( ( times_times @ real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( times_times @ rat @ X @ Y ) ) ) ).
% real_times_code
thf(fact_7314_single__valued__confluent,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A,Z3: A] :
( ( single_valued @ A @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ ( transitive_rtrancl @ A @ R3 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ).
% single_valued_confluent
thf(fact_7315_single__valued__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valued @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ B )] :
! [X4: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R )
=> ! [Z4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z4 ) @ R )
=> ( Y4 = Z4 ) ) ) ) ) ).
% single_valued_def
thf(fact_7316_single__valuedI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B,Z: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z ) @ R3 )
=> ( Y3 = Z ) ) )
=> ( single_valued @ A @ B @ R3 ) ) ).
% single_valuedI
thf(fact_7317_single__valuedD,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B ),X: A,Y: B,Z3: B] :
( ( single_valued @ A @ B @ R3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z3 ) @ R3 )
=> ( Y = Z3 ) ) ) ) ).
% single_valuedD
thf(fact_7318_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( single_valuedp @ A @ B
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R3 ) )
= ( single_valued @ A @ B @ R3 ) ) ).
% single_valuedp_single_valued_eq
thf(fact_7319_comm__monoid__set_Ozero__middle,axiom,
! [A: $tType,F3: A > A > A,Z3: A,P: nat,K2: nat,G3: nat > A,H: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P )
=> ( ( ord_less_eq @ nat @ K2 @ P )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [J: nat] : ( if @ A @ ( ord_less @ nat @ J @ K2 ) @ ( G3 @ J ) @ ( if @ A @ ( J = K2 ) @ Z3 @ ( H @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [J: nat] : ( if @ A @ ( ord_less @ nat @ J @ K2 ) @ ( G3 @ J ) @ ( H @ J ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.zero_middle
thf(fact_7320_comm__monoid__set_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ord @ B )
=> ! [F3: A > A > A,Z3: A,A3: B,C3: B,B2: B,D3: B,G3: B > A,H: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( A3 = C3 )
=> ( ( B2 = D3 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C3 @ X3 )
=> ( ( ord_less @ B @ X3 @ D3 )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z3 @ H @ ( set_or7035219750837199246ssThan @ B @ C3 @ D3 ) ) ) ) ) ) ) ) ).
% comm_monoid_set.ivl_cong
thf(fact_7321_comm__monoid__set_Ohead,axiom,
! [A: $tType,F3: A > A > A,Z3: A,M2: nat,N: nat,G3: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( F3 @ ( G3 @ M2 ) @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% comm_monoid_set.head
thf(fact_7322_comm__monoid__set_Olast__plus,axiom,
! [A: $tType,F3: A > A > A,Z3: A,M2: nat,N: nat,G3: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( F3 @ ( G3 @ N ) @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% comm_monoid_set.last_plus
thf(fact_7323_comm__monoid__set_Onat__ivl__Suc_H,axiom,
! [A: $tType,F3: A > A > A,Z3: A,M2: nat,N: nat,G3: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( F3 @ ( G3 @ ( suc @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% comm_monoid_set.nat_ivl_Suc'
thf(fact_7324_comm__monoid__set_OatLeast__Suc__atMost,axiom,
! [A: $tType,F3: A > A > A,Z3: A,M2: nat,N: nat,G3: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( F3 @ ( G3 @ M2 ) @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_atMost
thf(fact_7325_comm__monoid__set_OSuc__reindex__ivl,axiom,
! [A: $tType,F3: A > A > A,Z3: A,M2: nat,N: nat,G3: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G3 @ ( suc @ N ) ) )
= ( F3 @ ( G3 @ M2 )
@ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% comm_monoid_set.Suc_reindex_ivl
thf(fact_7326_comm__monoid__set_OatLeastLessThan__concat,axiom,
! [A: $tType,F3: A > A > A,Z3: A,M2: nat,N: nat,P: nat,G3: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P )
=> ( ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P ) ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_concat
thf(fact_7327_comm__monoid__set_OatLeastLessThan__Suc,axiom,
! [A: $tType,F3: A > A > A,Z3: A,A3: nat,B2: nat,G3: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G3 @ B2 ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_Suc
thf(fact_7328_comm__monoid__set_Onat__diff__reindex,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ ( minus_minus @ nat @ N @ ( suc @ I ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.nat_diff_reindex
thf(fact_7329_prod_Ocomm__monoid__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups778175481326437816id_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod.comm_monoid_set_axioms
thf(fact_7330_comm__monoid__mult__class_Oprod__def,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( groups_comm_monoid_F @ A @ B @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% comm_monoid_mult_class.prod_def
thf(fact_7331_comm__monoid__set_Ohead__if,axiom,
! [A: $tType,F3: A > A > A,Z3: A,N: nat,M2: nat,G3: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= Z3 ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% comm_monoid_set.head_if
thf(fact_7332_comm__monoid__set_Ocl__ivl__Suc,axiom,
! [A: $tType,F3: A > A > A,Z3: A,N: nat,M2: nat,G3: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= Z3 ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ) ).
% comm_monoid_set.cl_ivl_Suc
thf(fact_7333_comm__monoid__set_OatLeast__Suc__lessThan,axiom,
! [A: $tType,F3: A > A > A,Z3: A,M2: nat,N: nat,G3: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ord_less @ nat @ M2 @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( F3 @ ( G3 @ M2 ) @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_lessThan
thf(fact_7334_comm__monoid__set_Oop__ivl__Suc,axiom,
! [A: $tType,F3: A > A > A,Z3: A,N: nat,M2: nat,G3: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= Z3 ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% comm_monoid_set.op_ivl_Suc
thf(fact_7335_comm__monoid__set_OatMost__Suc,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_ord_atMost @ nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% comm_monoid_set.atMost_Suc
thf(fact_7336_comm__monoid__set_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_Suc_ivl
thf(fact_7337_comm__monoid__set_OlessThan__Suc,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( G3 @ N ) ) ) ) ).
% comm_monoid_set.lessThan_Suc
thf(fact_7338_comm__monoid__set_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_cl_Suc_ivl
thf(fact_7339_comm__monoid__set_Onested__swap_H,axiom,
! [A: $tType,F3: A > A > A,Z3: A,A3: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( A3 @ I ) @ ( set_ord_lessThan @ nat @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [J: nat] :
( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( A3 @ I @ J )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.nested_swap'
thf(fact_7340_comm__monoid__set_Onested__swap,axiom,
! [A: $tType,F3: A > A > A,Z3: A,A3: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( A3 @ I ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [J: nat] :
( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( A3 @ I @ J )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% comm_monoid_set.nested_swap
thf(fact_7341_comm__monoid__set_OatLeast1__atMost__eq,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.atLeast1_atMost_eq
thf(fact_7342_comm__monoid__set_OatMost__shift,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_ord_atMost @ nat @ N ) )
= ( F3 @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% comm_monoid_set.atMost_shift
thf(fact_7343_comm__monoid__set_OatLeast0__atMost__Suc,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% comm_monoid_set.atLeast0_atMost_Suc
thf(fact_7344_comm__monoid__set_OlessThan__Suc__shift,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( F3 @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% comm_monoid_set.lessThan_Suc_shift
thf(fact_7345_comm__monoid__set_OatLeast0__lessThan__Suc,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ N ) ) ) ) ).
% comm_monoid_set.atLeast0_lessThan_Suc
thf(fact_7346_comm__monoid__set_OatMost__Suc__shift,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( F3 @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ ( suc @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% comm_monoid_set.atMost_Suc_shift
thf(fact_7347_sum_Ocomm__monoid__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups778175481326437816id_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum.comm_monoid_set_axioms
thf(fact_7348_comm__monoid__add__class_Osum__def,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( groups_comm_monoid_F @ A @ B @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% comm_monoid_add_class.sum_def
thf(fact_7349_comm__monoid__set_Oshift__bounds__nat__ivl,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_nat_ivl
thf(fact_7350_comm__monoid__set_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_cl_nat_ivl
thf(fact_7351_comm__monoid__set_Oin__pairs__0,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( F3 @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% comm_monoid_set.in_pairs_0
thf(fact_7352_comm__monoid__set_OatLeastLessThan__rev,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat,M2: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_rev
thf(fact_7353_comm__monoid__set_OatLeastAtMost__rev,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat,M2: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_rev
thf(fact_7354_comm__monoid__set_Onat__group,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,K2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [M6: nat] : ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M6 @ K2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ M6 @ K2 ) @ K2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K2 ) ) ) ) ) ).
% comm_monoid_set.nat_group
thf(fact_7355_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat,M2: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_7356_comm__monoid__set_Oin__pairs,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( F3 @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.in_pairs
thf(fact_7357_comm__monoid__set_Oub__add__nat,axiom,
! [A: $tType,F3: A > A > A,Z3: A,M2: nat,N: nat,G3: nat > A,P: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P ) ) )
= ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P ) ) ) ) ) ) ) ).
% comm_monoid_set.ub_add_nat
thf(fact_7358_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_shift_0
thf(fact_7359_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
! [A: $tType,F3: A > A > A,Z3: A,M2: nat,N: nat,G3: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_shift_0
thf(fact_7360_comm__monoid__set_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ K2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_shift_bounds
thf(fact_7361_comm__monoid__set_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ K2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_shift_bounds
thf(fact_7362_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F3 @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% comm_monoid_set.atLeast0_lessThan_Suc_shift
thf(fact_7363_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F3 @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% comm_monoid_set.atLeast0_atMost_Suc_shift
thf(fact_7364_comm__monoid__set_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > A > A,Z3: A,H: nat > B,M2: nat,N: nat,G3: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( bij_betw @ nat @ B @ H @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H @ M2 ) @ ( H @ N ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ B @ ( H @ M2 ) @ ( H @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( comp @ B @ A @ nat @ G3 @ H ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_reindex
thf(fact_7365_comm__monoid__set_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > A > A,Z3: A,H: nat > B,M2: nat,N: nat,G3: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( bij_betw @ nat @ B @ H @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H @ M2 ) @ ( H @ N ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ B @ ( H @ M2 ) @ ( H @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( comp @ B @ A @ nat @ G3 @ H ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_reindex
thf(fact_7366_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_atMost_Suc_shift
thf(fact_7367_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_lessThan_Suc_shift
thf(fact_7368_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeast_atMost_pred_shift
thf(fact_7369_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeast_lessThan_pred_shift
thf(fact_7370_comm__monoid__set_Otriangle__reindex,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F3 @ Z3 @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I: nat,J: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [K3: nat] :
( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ I @ ( minus_minus @ nat @ K3 @ I ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.triangle_reindex
thf(fact_7371_comm__monoid__set_Otriangle__reindex__eq,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F3 @ Z3 @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I: nat,J: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ J ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [K3: nat] :
( groups_comm_monoid_F @ A @ nat @ F3 @ Z3
@ ^ [I: nat] : ( G3 @ I @ ( minus_minus @ nat @ K3 @ I ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% comm_monoid_set.triangle_reindex_eq
thf(fact_7372_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: int > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ int @ F3 @ Z3 @ G3 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( comp @ int @ A @ nat @ G3 @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeast_int_atMost_int_shift
thf(fact_7373_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType,F3: A > A > A,Z3: A,G3: int > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ int @ F3 @ Z3 @ G3 @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z3 @ ( comp @ int @ A @ nat @ G3 @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeast_int_lessThan_int_shift
thf(fact_7374_max__nat_Omonoid__axioms,axiom,
monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.monoid_axioms
thf(fact_7375_module__hom__compose__scale,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F3: A > real,C3: B] :
( ( vector_linear @ real @ A @ real @ ( real_V8093663219630862766scaleR @ A ) @ ( times_times @ real ) @ F3 )
=> ( real_Vector_linear @ A @ B
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ B @ ( F3 @ X4 ) @ C3 ) ) ) ) ).
% module_hom_compose_scale
thf(fact_7376_monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F3: A > A > A,Z3: A] :
( ( monoid @ A @ F3 @ Z3 )
=> ( semigroup @ A @ F3 ) ) ).
% monoid.axioms(1)
thf(fact_7377_monoid_Oright__neutral,axiom,
! [A: $tType,F3: A > A > A,Z3: A,A3: A] :
( ( monoid @ A @ F3 @ Z3 )
=> ( ( F3 @ A3 @ Z3 )
= A3 ) ) ).
% monoid.right_neutral
thf(fact_7378_monoid_Oleft__neutral,axiom,
! [A: $tType,F3: A > A > A,Z3: A,A3: A] :
( ( monoid @ A @ F3 @ Z3 )
=> ( ( F3 @ Z3 @ A3 )
= A3 ) ) ).
% monoid.left_neutral
thf(fact_7379_add_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.monoid_axioms
thf(fact_7380_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.monoid_axioms
thf(fact_7381_gcd__nat_Omonoid__axioms,axiom,
monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.monoid_axioms
thf(fact_7382_or_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.monoid_axioms
thf(fact_7383_xor_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% xor.monoid_axioms
thf(fact_7384_monoid_Ointro,axiom,
! [A: $tType,F3: A > A > A,Z3: A] :
( ( semigroup @ A @ F3 )
=> ( ( monoid_axioms @ A @ F3 @ Z3 )
=> ( monoid @ A @ F3 @ Z3 ) ) ) ).
% monoid.intro
thf(fact_7385_monoid__def,axiom,
! [A: $tType] :
( ( monoid @ A )
= ( ^ [F4: A > A > A,Z4: A] :
( ( semigroup @ A @ F4 )
& ( monoid_axioms @ A @ F4 @ Z4 ) ) ) ) ).
% monoid_def
thf(fact_7386_monoid__axioms_Ointro,axiom,
! [A: $tType,F3: A > A > A,Z3: A] :
( ! [A5: A] :
( ( F3 @ Z3 @ A5 )
= A5 )
=> ( ! [A5: A] :
( ( F3 @ A5 @ Z3 )
= A5 )
=> ( monoid_axioms @ A @ F3 @ Z3 ) ) ) ).
% monoid_axioms.intro
thf(fact_7387_monoid__axioms__def,axiom,
! [A: $tType] :
( ( monoid_axioms @ A )
= ( ^ [F4: A > A > A,Z4: A] :
( ! [A7: A] :
( ( F4 @ Z4 @ A7 )
= A7 )
& ! [A7: A] :
( ( F4 @ A7 @ Z4 )
= A7 ) ) ) ) ).
% monoid_axioms_def
thf(fact_7388_monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F3: A > A > A,Z3: A] :
( ( monoid @ A @ F3 @ Z3 )
=> ( monoid_axioms @ A @ F3 @ Z3 ) ) ).
% monoid.axioms(2)
thf(fact_7389_Enum_Ortranclp__rtrancl__eq,axiom,
! [A: $tType] :
( ( transitive_rtranclp @ A )
= ( ^ [R: A > A > $o,X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ ( transitive_rtrancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R ) ) ) ) ) ) ).
% Enum.rtranclp_rtrancl_eq
thf(fact_7390_rtrancl__def,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_rtranclp @ A
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R ) ) ) ) ) ) ).
% rtrancl_def
thf(fact_7391_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_rtranclp @ A
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) )
= ( ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% Transitive_Closure.rtranclp_rtrancl_eq
thf(fact_7392_converse__rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R3: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P2: A > B > $o] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( ( R3 @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Aa2 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P2 @ Aa2 @ Ba )
=> ( P2 @ A5 @ B4 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtranclp_induct2
thf(fact_7393_converse__rtranclpE2,axiom,
! [A: $tType,B: $tType,R3: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Xa2: A,Xb: B,Za: A,Zb: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ Za @ Zb ) )
=> ( ( ( product_Pair @ A @ B @ Xa2 @ Xb )
!= ( product_Pair @ A @ B @ Za @ Zb ) )
=> ~ ! [A5: A,B4: B] :
( ( R3 @ ( product_Pair @ A @ B @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ~ ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) ) ) ) ).
% converse_rtranclpE2
thf(fact_7394_rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R3: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P2: A > B > $o] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R3 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( ( R3 @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( P2 @ A5 @ B4 )
=> ( P2 @ Aa2 @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtranclp_induct2
thf(fact_7395_Card__order__infinite__not__under,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ~ ? [A9: A] :
( ( field2 @ A @ R3 )
= ( order_under @ A @ R3 @ A9 ) ) ) ) ).
% Card_order_infinite_not_under
thf(fact_7396_card__order__on__Card__order,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A6 @ R3 )
=> ( ( A6
= ( field2 @ A @ R3 ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) ) ) ).
% card_order_on_Card_order
thf(fact_7397_card__order__on,axiom,
! [A: $tType,A6: set @ A] :
? [X_12: set @ ( product_prod @ A @ A )] : ( bNF_Ca8970107618336181345der_on @ A @ A6 @ X_12 ) ).
% card_order_on
thf(fact_7398_card__order__on__well__order__on,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A6 @ R3 )
=> ( order_well_order_on @ A @ A6 @ R3 ) ) ).
% card_order_on_well_order_on
thf(fact_7399_natLeq__Card__order,axiom,
bNF_Ca8970107618336181345der_on @ nat @ ( field2 @ nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).
% natLeq_Card_order
thf(fact_7400_infinite__Card__order__limit,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( A3 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X3 ) @ R3 ) ) ) ) ) ).
% infinite_Card_order_limit
thf(fact_7401_Card__order__trans,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A,Z3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( X != Y )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( Y != Z3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ R3 )
=> ( ( X != Z3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ R3 ) ) ) ) ) ) ) ).
% Card_order_trans
thf(fact_7402_exists__isCardSuc,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ? [X_12: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] : ( bNF_Ca6246979054910435723ardSuc @ A @ R3 @ X_12 ) ) ).
% exists_isCardSuc
thf(fact_7403_cardSuc__finite,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( finite_finite2 @ ( set @ A ) @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) )
= ( finite_finite2 @ A @ ( field2 @ A @ R3 ) ) ) ) ).
% cardSuc_finite
thf(fact_7404_cardSuc__isCardSuc,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( bNF_Ca6246979054910435723ardSuc @ A @ R3 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) ) ).
% cardSuc_isCardSuc
thf(fact_7405_cardSuc__Card__order,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) ) ).
% cardSuc_Card_order
thf(fact_7406_cardSuc__def,axiom,
! [A: $tType] :
( ( bNF_Ca8387033319878233205ardSuc @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] : ( fChoice @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca6246979054910435723ardSuc @ A @ R ) ) ) ) ).
% cardSuc_def
thf(fact_7407_infinite__cardSuc__regularCard,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( bNF_Ca7133664381575040944arCard @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) ) ) ).
% infinite_cardSuc_regularCard
thf(fact_7408_Cinfinite__limit__finite,axiom,
! [A: $tType,X8: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X8 )
=> ( ( ord_less_eq @ ( set @ A ) @ X8 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ! [Xa: A] :
( ( member @ A @ Xa @ X8 )
=> ( ( Xa != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
thf(fact_7409_Cinfinite__limit,axiom,
! [A: $tType,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( X != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X3 ) @ R3 ) ) ) ) ).
% Cinfinite_limit
thf(fact_7410_Cinfinite__limit2,axiom,
! [A: $tType,X1: A,R3: set @ ( product_prod @ A @ A ),X2: A] :
( ( member @ A @ X1 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ X2 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( X1 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X3 ) @ R3 )
& ( X2 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X3 ) @ R3 ) ) ) ) ) ).
% Cinfinite_limit2
thf(fact_7411_card__of__cardSuc__finite,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ ( set @ A ) @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) ) )
= ( finite_finite2 @ A @ A6 ) ) ).
% card_of_cardSuc_finite
thf(fact_7412_cardSuc__UNION,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),As3: ( set @ A ) > ( set @ B ),B6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) @ As3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As3 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) ) ) )
=> ( ( 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 @ B6 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) )
& ( ord_less_eq @ ( set @ B ) @ B6 @ ( As3 @ X3 ) ) ) ) ) ) ) ) ).
% cardSuc_UNION
thf(fact_7413_card__of__Card__order,axiom,
! [A: $tType,A6: set @ A] : ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) ).
% card_of_Card_order
thf(fact_7414_card__of__card__order__on,axiom,
! [A: $tType,A6: set @ A] : ( bNF_Ca8970107618336181345der_on @ A @ A6 @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) ).
% card_of_card_order_on
thf(fact_7415_card__of__UNION__ordLeq__infinite__Field,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),I5: set @ B,A6: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A6 @ X3 ) ) @ R3 ) @ ( 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 ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite_Field
thf(fact_7416_Card__order__Times__same__infinite,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( 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 @ R3 )
@ ^ [Uu3: A] : ( field2 @ A @ R3 ) ) )
@ R3 )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ A ) @ A ) ) ) ) ).
% Card_order_Times_same_infinite
thf(fact_7417_card__of__Sigma__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),I5: set @ B,A6: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A6 @ X3 ) ) @ R3 ) @ ( 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 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite_Field
thf(fact_7418_card__of__Times__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A6: set @ B,B6: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( 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 ) @ R3 ) @ ( 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 @ B6 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( 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] : B6 ) )
@ R3 )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Times_ordLeq_infinite_Field
thf(fact_7419_Card__order__iff__ordLeq__card__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( 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 ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R3 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% Card_order_iff_ordLeq_card_of
thf(fact_7420_cardSuc__mono__ordLeq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( 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 @ R3 ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R5 ) ) @ ( 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 ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% cardSuc_mono_ordLeq
thf(fact_7421_card__order__on__def,axiom,
! [A: $tType] :
( ( bNF_Ca8970107618336181345der_on @ A )
= ( ^ [A8: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A8 @ R )
& ! [R7: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A8 @ R7 )
=> ( 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 ) ) @ R @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% card_order_on_def
thf(fact_7422_card__of__empty1,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
| ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( 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 ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_empty1
thf(fact_7423_card__of__Un__ordLeq__infinite__Field,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A6: set @ B,B6: set @ B] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( 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 ) @ R3 ) @ ( 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 @ B6 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( 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 @ B6 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ) ).
% card_of_Un_ordLeq_infinite_Field
thf(fact_7424_exists__minim__Card__order,axiom,
! [A: $tType,R2: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R2
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R2 )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ X3 ) @ X3 ) )
=> ? [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R2 )
& ! [Xa: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa @ 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 ) ) @ X3 @ Xa ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Card_order
thf(fact_7425_Card__order__empty,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( 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 ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% Card_order_empty
thf(fact_7426_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),B2: B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ A @ R3 )
!= ( 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 @ ( insert @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% Card_order_singl_ordLeq
thf(fact_7427_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) ) ) @ R3
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu3: B] : ( field2 @ A @ R3 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ) ) ).
% Card_order_Times2
thf(fact_7428_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),B6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( B6
!= ( 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 ) ) ) @ R3
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R3 )
@ ^ [Uu3: A] : B6 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ B ) ) ) ) ) ).
% Card_order_Times1
thf(fact_7429_card__of__Plus__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A6: set @ B,B6: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( 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 ) @ R3 ) @ ( 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 @ B6 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( 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 @ B6 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLeq_infinite_Field
thf(fact_7430_Card__order__Plus1,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),B6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( 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 ) ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R3 ) @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% Card_order_Plus1
thf(fact_7431_Card__order__Plus2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( 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 ) ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A6 @ ( field2 @ A @ R3 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% Card_order_Plus2
thf(fact_7432_card__of__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A6: set @ A,B6: set @ B,C5: 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 @ B6 ) ) @ ( 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 @ C5 @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C5 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_mono2
thf(fact_7433_card__of__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A6: set @ A,B6: set @ B,C5: 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 @ B6 ) ) @ ( 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 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B6 @ C5 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_mono1
thf(fact_7434_card__of__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A6: set @ A,B6: set @ B,C5: set @ C,D5: 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 @ B6 ) ) @ ( 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 @ C5 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D5 ) ) @ ( 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 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B6 @ D5 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_mono
thf(fact_7435_card__of__Plus2,axiom,
! [B: $tType,A: $tType,B6: 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 @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A6 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus2
thf(fact_7436_card__of__Plus1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: 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 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus1
thf(fact_7437_card__of__Un__Plus__ordLeq,axiom,
! [A: $tType,A6: set @ A,B6: 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 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A6 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% card_of_Un_Plus_ordLeq
thf(fact_7438_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_7439_card__of__Times1,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: 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 @ B6 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B6
@ ^ [Uu3: B] : A6 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% card_of_Times1
thf(fact_7440_card__of__Times2,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: 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 @ B6 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% card_of_Times2
thf(fact_7441_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 @ ( insert @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_singl_ordLeq
thf(fact_7442_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_7443_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_7444_card__of__least,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A6 @ R3 )
=> ( 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 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_least
thf(fact_7445_card__of__ordLeq__infinite,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: 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 @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( finite_finite2 @ A @ A6 )
=> ~ ( finite_finite2 @ B @ B6 ) ) ) ).
% card_of_ordLeq_infinite
thf(fact_7446_card__of__ordLeq__finite,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: 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 @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( finite_finite2 @ B @ B6 )
=> ( finite_finite2 @ A @ A6 ) ) ) ).
% card_of_ordLeq_finite
thf(fact_7447_card__of__Sigma__ordLeq__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,B6: set @ A,I5: set @ B,A6: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B6 )
=> ( ( 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 @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A6 @ X3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( 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 @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite
thf(fact_7448_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_7449_card__of__Sigma__mono1,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,A6: A > ( set @ B ),B6: A > ( set @ C )] :
( ! [X3: A] :
( ( member @ A @ X3 @ 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 @ X3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( B6 @ X3 ) ) ) @ ( 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 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Sigma_mono1
thf(fact_7450_card__of__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A6: set @ A,B6: set @ B,C5: 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 @ B6 ) ) @ ( 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] : C5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ B6
@ ^ [Uu3: B] : C5 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% card_of_Times_mono1
thf(fact_7451_card__of__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A6: set @ A,B6: set @ B,C5: 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 @ B6 ) ) @ ( 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 @ C5
@ ^ [Uu3: C] : A6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ C5
@ ^ [Uu3: C] : B6 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% card_of_Times_mono2
thf(fact_7452_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_7453_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_7454_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,B6: set @ A,I5: set @ B,A6: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B6 )
=> ( ( 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 @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A6 @ X3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( 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 @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite
thf(fact_7455_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [G4: B > A] :
( ( image2 @ B @ A @ G4 @ B6 )
= 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 @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeq2
thf(fact_7456_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_7457_card__of__ordLeqI,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,B6: set @ B] :
( ( inj_on @ A @ B @ F3 @ A6 )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( member @ B @ ( F3 @ A5 ) @ B6 ) )
=> ( 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_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeqI
thf(fact_7458_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A6 ) @ B6 ) ) )
= ( 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_ordLeq @ A @ B ) ) ) ).
% card_of_ordLeq
thf(fact_7459_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,B6: set @ A,F3: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( 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 @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% surj_imp_ordLeq
thf(fact_7460_card__of__mono1,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( 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 @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_mono1
thf(fact_7461_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A12: A,A23: A,A6: set @ A,B14: B,B23: B,B6: set @ B] :
( ( ( A12 != A23 )
& ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A12 @ ( insert @ A @ A23 @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 ) )
=> ( ( ( B14 != B23 )
& ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ B14 @ ( insert @ B @ B23 @ ( bot_bot @ ( set @ B ) ) ) ) @ B6 ) )
=> ( 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 @ B6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times
thf(fact_7462_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A12: A,A23: A,A6: set @ A,B6: set @ B] :
( ( ( A12 != A23 )
& ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A12 @ ( insert @ A @ A23 @ ( 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 @ B6 ) ) @ ( 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 @ B6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu3: A] : B6 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times_aux
thf(fact_7463_card__of__well__order__on,axiom,
! [A: $tType,A6: set @ A] : ( order_well_order_on @ A @ A6 @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) ).
% card_of_well_order_on
thf(fact_7464_Field__card__of,axiom,
! [A: $tType,A6: set @ A] :
( ( field2 @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) )
= A6 ) ).
% Field_card_of
thf(fact_7465_card__of__Well__order,axiom,
! [A: $tType,A6: set @ A] : ( order_well_order_on @ A @ ( field2 @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) ).
% card_of_Well_order
thf(fact_7466_ordLeq__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),R5: 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 ) ) @ R3 @ R5 ) @ ( 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 @ R3 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ A6
@ ^ [Uu3: C] : ( field2 @ B @ R5 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% ordLeq_Times_mono2
thf(fact_7467_ordLeq__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),C5: 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 ) ) @ R3 @ R5 ) @ ( 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 @ R3 )
@ ^ [Uu3: A] : C5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ ( field2 @ B @ R5 )
@ ^ [Uu3: B] : C5 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% ordLeq_Times_mono1
thf(fact_7468_card__of__mono2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: 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 ) ) @ R3 @ R5 ) @ ( 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 @ R3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R5 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_mono2
thf(fact_7469_card__of__Field__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( 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 @ R3 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_Field_ordLess
thf(fact_7470_ordLeq__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),P: set @ ( product_prod @ C @ C ),P10: 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 ) ) @ R3 @ R5 ) @ ( 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 ) ) @ P @ P10 ) @ ( 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 @ R3 ) @ ( field2 @ C @ P ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R5 ) @ ( field2 @ D @ P10 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordLeq_Plus_mono
thf(fact_7471_ordLeq__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),C5: 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 ) ) @ R3 @ R5 ) @ ( 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 @ R3 ) @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R5 ) @ C5 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordLeq_Plus_mono1
thf(fact_7472_ordLeq__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),R5: 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 ) ) @ R3 @ R5 ) @ ( 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 @ R3 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A6 @ ( field2 @ B @ R5 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordLeq_Plus_mono2
thf(fact_7473_cardSuc__ordLeq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( 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 ) ) ) @ R3 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( set @ A ) ) ) ) ).
% cardSuc_ordLeq
thf(fact_7474_card__of__def,axiom,
! [A: $tType] :
( ( bNF_Ca6860139660246222851ard_of @ A )
= ( ^ [A8: set @ A] : ( fChoice @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca8970107618336181345der_on @ A @ A8 ) ) ) ) ).
% card_of_def
thf(fact_7475_regularCard__UNION,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),As3: A > ( set @ B ),B6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca7133664381575040944arCard @ A @ R3 )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ ( set @ B ) @ R3 @ As3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ As3 @ ( field2 @ A @ R3 ) ) ) )
=> ( ( 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 @ B6 ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( ord_less_eq @ ( set @ B ) @ B6 @ ( As3 @ X3 ) ) ) ) ) ) ) ) ).
% regularCard_UNION
thf(fact_7476_Card__order__iff__Restr__underS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R3 ) )
=> ( 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 ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R3 @ X4 )
@ ^ [Uu3: A] : ( order_underS @ A @ R3 @ X4 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R3 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% Card_order_iff_Restr_underS
thf(fact_7477_card__of__Plus__ordLess__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,C5: set @ A,A6: set @ B,B6: set @ C] :
( ~ ( finite_finite2 @ A @ C5 )
=> ( ( 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 @ C5 ) ) @ ( 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 @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( 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 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite
thf(fact_7478_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_7479_card__of__Plus__ordLess__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A6: set @ B,B6: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) @ R3 ) @ ( 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 @ B6 ) @ R3 ) @ ( 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 @ B6 ) ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite_Field
thf(fact_7480_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: 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 ) ) @ R3 @ R5 ) @ ( 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 @ R3 ) ) @ R5 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ).
% BNF_Cardinal_Order_Relation.ordLess_Field
thf(fact_7481_Card__order__Pow,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( 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 ) ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow2 @ A @ ( field2 @ A @ R3 ) ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% Card_order_Pow
thf(fact_7482_cardSuc__greater,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( 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 ) ) ) @ R3 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% cardSuc_greater
thf(fact_7483_finite__iff__ordLess__natLeq,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A8: 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 @ A8 ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ A @ nat ) ) ) ) ).
% finite_iff_ordLess_natLeq
thf(fact_7484_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B6: set @ A,A6: set @ B] :
( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ? [F4: B > A] :
( ( image2 @ B @ A @ F4 @ A6 )
= B6 ) )
= ( 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 @ B6 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% card_of_ordLess2
thf(fact_7485_cardSuc__least,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( 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 ) ) @ R3 @ R5 ) @ ( 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 @ R3 ) @ R5 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_least
thf(fact_7486_cardSuc__ordLess__ordLeq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( 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 ) ) @ R3 @ R5 ) @ ( 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 @ R3 ) @ R5 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_ordLess_ordLeq
thf(fact_7487_isCardSuc__def,axiom,
! [A: $tType] :
( ( bNF_Ca6246979054910435723ardSuc @ A )
= ( ^ [R: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R7 ) @ R7 )
& ( 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 ) ) ) @ R @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
& ! [R8: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R8 ) @ R8 )
& ( 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 ) ) ) @ R @ R8 ) @ ( 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 ) ) ) @ R7 @ R8 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) ).
% isCardSuc_def
thf(fact_7488_card__of__ordLess,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
( ( ~ ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A6 ) @ B6 ) ) )
= ( 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 @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ).
% card_of_ordLess
thf(fact_7489_cardSuc__ordLeq__ordLess,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( 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 ) ) ) @ R5 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) @ ( 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 ) ) @ R5 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% cardSuc_ordLeq_ordLess
thf(fact_7490_card__of__underS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( 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 @ R3 @ A3 ) ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% card_of_underS
thf(fact_7491_cardSuc__least__aux,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R5 ) @ R5 )
=> ( ( 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 ) ) ) @ R3 @ R5 ) @ ( 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 @ R3 ) @ R5 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ).
% cardSuc_least_aux
thf(fact_7492_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),P: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ B @ P )
!= ( 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 ) ) @ P @ R3 ) @ ( 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 @ R3 )
@ ^ [Uu3: A] : ( field2 @ B @ P ) ) )
@ R3 )
@ ( 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 @ P )
@ ^ [Uu3: B] : ( field2 @ A @ R3 ) ) )
@ R3 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ) ).
% Card_order_Times_infinite
thf(fact_7493_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( B6
!= ( 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 @ B6 ) @ ( 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] : B6 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(2)
thf(fact_7494_cardSuc__invar__ordIso,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 )
=> ( ( 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 @ R3 ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R5 ) ) @ ( 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 ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% cardSuc_invar_ordIso
thf(fact_7495_card__order__on__ordIso,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A6 @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ A6 @ 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 ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ).
% card_order_on_ordIso
thf(fact_7496_card__of__Times__Plus__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,A6: set @ A,B6: set @ B,C5: 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 @ B6 @ C5 ) ) )
@ ( 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] : B6 )
@ ( product_Sigma @ A @ C @ A6
@ ^ [Uu3: A] : C5 ) ) ) )
@ ( 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_7497_card__of__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,A6: set @ A,B6: set @ B,C5: 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 @ B6 ) ) @ ( 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 @ C5 @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C5 @ B6 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_cong2
thf(fact_7498_card__of__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,A6: set @ A,B6: set @ B,C5: 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 @ B6 ) ) @ ( 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 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B6 @ C5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_cong1
thf(fact_7499_card__of__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A6: set @ A,B6: set @ B,C5: set @ C,D5: 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 @ B6 ) ) @ ( 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 @ C5 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D5 ) ) @ ( 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 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B6 @ D5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_cong
thf(fact_7500_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_7501_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_7502_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_7503_card__of__ordIso__finite,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: 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 @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ A6 )
= ( finite_finite2 @ B @ B6 ) ) ) ).
% card_of_ordIso_finite
thf(fact_7504_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_7505_ordIso__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),P: set @ ( product_prod @ C @ C ),P10: 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 ) ) @ R3 @ R5 ) @ ( 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 ) ) @ P @ P10 ) @ ( 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 @ R3 ) @ ( field2 @ C @ P ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R5 ) @ ( field2 @ D @ P10 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordIso_Plus_cong
thf(fact_7506_ordIso__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B ),C5: 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 ) ) @ R3 @ R5 ) @ ( 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 @ R3 ) @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R5 ) @ C5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordIso_Plus_cong1
thf(fact_7507_ordIso__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),R5: 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 ) ) @ R3 @ R5 ) @ ( 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 @ R3 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A6 @ ( field2 @ B @ R5 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordIso_Plus_cong2
thf(fact_7508_card__of__cong,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R5: 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 ) ) @ R3 @ R5 ) @ ( 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 @ R3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R5 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_cong
thf(fact_7509_Card__order__ordIso2,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) ) @ R3 @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 ) ) ) ).
% Card_order_ordIso2
thf(fact_7510_Card__order__ordIso,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) ) @ R5 @ R3 ) @ ( bNF_Wellorder_ordIso @ B @ A ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 ) ) ) ).
% Card_order_ordIso
thf(fact_7511_card__of__unique,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A6 @ R3 )
=> ( 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 ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_unique
thf(fact_7512_card__of__ordIso,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ( ? [F4: A > B] : ( bij_betw @ A @ B @ F4 @ A6 @ B6 ) )
= ( 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 ) ) ) ).
% card_of_ordIso
thf(fact_7513_card__of__ordIsoI,axiom,
! [B: $tType,A: $tType,F3: A > B,A6: set @ A,B6: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A6 @ B6 )
=> ( 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 ) ) ) ).
% card_of_ordIsoI
thf(fact_7514_card__of__Times__commute,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: 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] : B6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B6
@ ^ [Uu3: B] : A6 ) ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% card_of_Times_commute
thf(fact_7515_internalize__card__of__ordLeq2,axiom,
! [A: $tType,B: $tType,A6: set @ A,C5: 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 @ C5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B8 @ C5 )
& ( 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 @ B8 ) ) @ ( 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 @ B8 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq2
thf(fact_7516_ordIso__card__of__imp__Card__order,axiom,
! [B: $tType,A: $tType,R3: 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 ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ B @ A6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) ) ).
% ordIso_card_of_imp_Card_order
thf(fact_7517_Card__order__iff__ordIso__card__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( 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 ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R3 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% Card_order_iff_ordIso_card_of
thf(fact_7518_card__of__Field__ordIso,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( 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 @ R3 ) ) @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_Field_ordIso
thf(fact_7519_internalize__card__of__ordLeq,axiom,
! [A: $tType,B: $tType,A6: set @ A,R3: 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 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B8 @ ( field2 @ B @ R3 ) )
& ( 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 @ B8 ) ) @ ( 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 @ B8 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq
thf(fact_7520_card__of__ordIso__finite__Field,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ B @ A6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
= ( finite_finite2 @ B @ A6 ) ) ) ) ).
% card_of_ordIso_finite_Field
thf(fact_7521_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_7522_regularCard__def,axiom,
! [A: $tType] :
( ( bNF_Ca7133664381575040944arCard @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [K6: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ K6 @ ( field2 @ A @ R ) )
& ( bNF_Ca7293521722713021262ofinal @ A @ K6 @ 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 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ K6 ) @ R ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% regularCard_def
thf(fact_7523_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( B6
!= ( 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 @ B6 ) @ ( 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 @ B6
@ ^ [Uu3: B] : A6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(3)
thf(fact_7524_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( B6
!= ( 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 @ B6 ) @ ( 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] : B6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(1)
thf(fact_7525_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( B6
!= ( 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 @ B6 ) @ ( 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] : B6 ) )
@ ( 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 @ B6
@ ^ [Uu3: B] : A6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ).
% card_of_Times_infinite
thf(fact_7526_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( B6
!= ( 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 @ B6 ) @ ( 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 @ B6
@ ^ [Uu3: B] : A6 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ B @ A ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(4)
thf(fact_7527_Card__order__Plus__infinite,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),P: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) ) @ P @ R3 ) @ ( 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 @ R3 ) @ ( field2 @ B @ P ) ) ) @ R3 ) @ ( 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 @ P ) @ ( field2 @ A @ R3 ) ) ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ) ).
% Card_order_Plus_infinite
thf(fact_7528_card__of__Plus__infinite,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: 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 @ B6 ) @ ( 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 @ B6 ) ) @ ( 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 @ B6 @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Plus_infinite
thf(fact_7529_card__of__Plus__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,A6: set @ A,B6: set @ B,C5: 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 @ B6 ) @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_Plus @ A @ ( sum_sum @ B @ C ) @ A6 @ ( sum_Plus @ B @ C @ B6 @ C5 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_assoc
thf(fact_7530_card__of__bool,axiom,
! [A: $tType,A12: A,A23: A] :
( ( A12 != A23 )
=> ( 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 @ ( insert @ A @ A12 @ ( insert @ A @ A23 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ) ).
% card_of_bool
thf(fact_7531_card__of__Plus__commute,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: 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 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B6 @ A6 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_commute
thf(fact_7532_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_7533_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_7534_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_7535_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_7536_card__of__Plus__infinite2,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: 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 @ B6 ) @ ( 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 @ B6 @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ).
% card_of_Plus_infinite2
thf(fact_7537_card__of__Plus__infinite1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: 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 @ B6 ) @ ( 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 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A6 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% card_of_Plus_infinite1
thf(fact_7538_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_7539_card__of__Func__UNIV,axiom,
! [B: $tType,A: $tType,B6: 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 ) ) @ B6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F4: A > B] : ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) @ B6 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > B ) @ ( A > B ) ) ) ).
% card_of_Func_UNIV
thf(fact_7540_Func__Times__Range,axiom,
! [C: $tType,B: $tType,A: $tType,A6: set @ A,B6: set @ B,C5: 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 @ B6
@ ^ [Uu3: B] : C5 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ ( A > B ) @ ( A > C ) )
@ ( product_Sigma @ ( A > B ) @ ( A > C ) @ ( bNF_Wellorder_Func @ A @ B @ A6 @ B6 )
@ ^ [Uu3: A > B] : ( bNF_Wellorder_Func @ A @ C @ A6 @ C5 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% Func_Times_Range
thf(fact_7541_card__of__Func__Times,axiom,
! [C: $tType,B: $tType,A: $tType,A6: set @ A,B6: set @ B,C5: 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] : B6 )
@ C5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B > C ) @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A6 @ ( bNF_Wellorder_Func @ B @ C @ B6 @ C5 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( ( product_prod @ A @ B ) > C ) @ ( A > B > C ) ) ) ).
% card_of_Func_Times
% Type constructors (861)
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1,axiom,
! [A10: $tType,A19: $tType] :
( ( ( finite_finite @ A10 )
& ( finite_finite @ A19 ) )
=> ( finite_finite @ ( product_prod @ A10 @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_2,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( finite_finite @ ( option @ A10 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_3,axiom,
! [A10: $tType,A19: $tType] :
( ( ( finite_finite @ A10 )
& ( finite_finite @ A19 ) )
=> ( finite_finite @ ( sum_sum @ A10 @ A19 ) ) ) ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_4,axiom,
finite_finite @ char ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_5,axiom,
finite_finite @ $o ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_6,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( finite_finite @ ( set @ A10 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite_7,axiom,
! [A10: $tType,A19: $tType] :
( ( ( finite_finite @ A10 )
& ( finite_finite @ A19 ) )
=> ( finite_finite @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,axiom,
! [A10: $tType,A19: $tType] :
( ( comple592849572758109894attice @ A19 )
=> ( counta4013691401010221786attice @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A10: $tType,A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( condit1219197933456340205attice @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
! [A10: $tType,A19: $tType] :
( ( counta3822494911875563373attice @ A19 )
=> ( counta3822494911875563373attice @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A10: $tType,A19: $tType] :
( ( comple592849572758109894attice @ A19 )
=> ( comple592849572758109894attice @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A10: $tType,A19: $tType] :
( ( bounded_lattice @ A19 )
=> ( bounde4967611905675639751up_bot @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A10: $tType,A19: $tType] :
( ( bounded_lattice @ A19 )
=> ( bounde4346867609351753570nf_top @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A10: $tType,A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( comple6319245703460814977attice @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A10: $tType,A19: $tType] :
( ( boolea8198339166811842893lgebra @ A19 )
=> ( boolea8198339166811842893lgebra @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A10: $tType,A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( comple9053668089753744459l_ccpo @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A10: $tType,A19: $tType] :
( ( semilattice_sup @ A19 )
=> ( semilattice_sup @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A10: $tType,A19: $tType] :
( ( semilattice_inf @ A19 )
=> ( semilattice_inf @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A10: $tType,A19: $tType] :
( ( bounded_lattice @ A19 )
=> ( bounded_lattice @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A10: $tType,A19: $tType] :
( ( order_top @ A19 )
=> ( order_top @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A10: $tType,A19: $tType] :
( ( order_bot @ A19 )
=> ( order_bot @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A10: $tType,A19: $tType] :
( ( ( finite_finite @ A10 )
& ( countable @ A19 ) )
=> ( countable @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A10: $tType,A19: $tType] :
( ( preorder @ A19 )
=> ( preorder @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A10: $tType,A19: $tType] :
( ( lattice @ A19 )
=> ( lattice @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A10: $tType,A19: $tType] :
( ( order @ A19 )
=> ( order @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A10: $tType,A19: $tType] :
( ( top @ A19 )
=> ( top @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A10: $tType,A19: $tType] :
( ( ord @ A19 )
=> ( ord @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A10: $tType,A19: $tType] :
( ( bot @ A19 )
=> ( bot @ ( A10 > A19 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A10: $tType,A19: $tType] :
( ( uminus @ A19 )
=> ( uminus @ ( A10 > A19 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_8,axiom,
condit1219197933456340205attice @ int ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Rings_Onormalization__semidom__multiplicative,axiom,
normal6328177297339901930cative @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
topolo4958980785337419405_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
topolo1944317154257567458pology @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
topolo5987344860129210374id_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
topolo2564578578187576103pology @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri2026040879449505780visors @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
topolo4211221413907600880p_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
semido2269285787275462019factor @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
semiri6843258321239162965malize @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__mult,axiom,
topolo1898628316856586783d_mult @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
topolo6943815403480290642id_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s4317794764714335236cancel @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
topological_t2_space @ int ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
normal8620421768224518004emidom @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_9,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_10,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
semiring_1_cancel @ int ).
thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Countable_Ocountable_11,axiom,
countable @ int ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_12,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
idom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Oidom__divide,axiom,
idom_divide @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0 @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Lattices_Olattice_13,axiom,
lattice @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
semiring_gcd @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
semiring_Gcd @ int ).
thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
mult_zero @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_14,axiom,
order @ int ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int ).
thf(tcon_Int_Oint___Rings_Osemidom,axiom,
semidom @ int ).
thf(tcon_Int_Oint___Orderings_Oord_15,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_16,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___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___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_17,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_18,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_19,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_20,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_21,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_22,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_23,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_24,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_25,axiom,
normal6328177297339901930cative @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_26,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_27,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_28,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_29,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_30,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_31,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_32,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_33,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_34,axiom,
ordere8940638589300402666id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni5634975068530333245id_add @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_35,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_36,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_37,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_38,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_39,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_40,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_41,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_42,axiom,
topolo4211221413907600880p_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_43,axiom,
semido2269285787275462019factor @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_44,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_45,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_46,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_47,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__mult_48,axiom,
topolo1898628316856586783d_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_49,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_50,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_51,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_52,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_53,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_54,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_55,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_56,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom_57,axiom,
normal8620421768224518004emidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_58,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_59,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_60,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_61,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_62,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_63,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_64,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_65,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_66,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_67,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_68,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_69,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_70,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_71,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_72,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_73,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_74,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_75,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_76,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_77,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_78,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_79,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_80,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_81,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_82,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_83,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Countable_Ocountable_84,axiom,
countable @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_85,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_86,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_87,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_88,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_89,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_90,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_91,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_92,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_93,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_94,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_95,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_96,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_97,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_98,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom_99,axiom,
semidom @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_100,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_101,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Power_Opower_102,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_103,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_104,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_105,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_106,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_107,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_108,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_109,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_110,axiom,
ord @ num ).
thf(tcon_Num_Onum___Nat_Osize_111,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_112,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_113,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_114,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_115,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_116,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_117,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_118,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_119,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_120,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_121,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_122,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_123,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_124,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_125,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_126,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_127,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_128,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_129,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_130,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_131,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_132,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_133,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_134,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_135,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_136,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_137,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_138,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_139,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_140,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_141,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_142,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_143,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_144,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_145,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_146,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_147,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_148,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_149,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_150,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_151,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_152,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_153,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_154,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_155,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_156,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_157,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_158,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_159,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_160,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_161,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_162,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Countable_Ocountable_163,axiom,
countable @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__neq__one_164,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_165,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_166,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_167,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_168,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_169,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_170,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_171,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_172,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_173,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_174,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_175,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_176,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_177,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_178,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_179,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_180,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_181,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_182,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_183,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_184,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_185,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_186,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_187,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_188,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_189,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_190,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_191,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_192,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_193,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_194,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_195,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_196,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_197,axiom,
! [A10: $tType] : ( counta4013691401010221786attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_198,axiom,
! [A10: $tType] : ( condit1219197933456340205attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_199,axiom,
! [A10: $tType] : ( counta3822494911875563373attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_200,axiom,
! [A10: $tType] : ( comple592849572758109894attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_201,axiom,
! [A10: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_202,axiom,
! [A10: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_203,axiom,
! [A10: $tType] : ( comple6319245703460814977attice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_204,axiom,
! [A10: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_205,axiom,
! [A10: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_206,axiom,
! [A10: $tType] : ( semilattice_sup @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_207,axiom,
! [A10: $tType] : ( semilattice_inf @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_208,axiom,
! [A10: $tType] : ( bounded_lattice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_209,axiom,
! [A10: $tType] : ( order_top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_210,axiom,
! [A10: $tType] : ( order_bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_211,axiom,
! [A10: $tType] :
( ( finite_finite @ A10 )
=> ( countable @ ( set @ A10 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_212,axiom,
! [A10: $tType] : ( preorder @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_213,axiom,
! [A10: $tType] : ( lattice @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_214,axiom,
! [A10: $tType] : ( order @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_215,axiom,
! [A10: $tType] : ( top @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_216,axiom,
! [A10: $tType] : ( ord @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_217,axiom,
! [A10: $tType] : ( bot @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_218,axiom,
! [A10: $tType] : ( uminus @ ( set @ A10 ) ) ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_219,axiom,
counta4013691401010221786attice @ $o ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_220,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_221,axiom,
counta3822494911875563373attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_222,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_223,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_224,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_225,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_226,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_227,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_228,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_229,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_230,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_231,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_232,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_233,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_234,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_235,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_236,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Countable_Ocountable_237,axiom,
countable @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_238,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_239,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_240,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_241,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_242,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_243,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_244,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_245,axiom,
uminus @ $o ).
thf(tcon_List_Olist___Countable_Ocountable_246,axiom,
! [A10: $tType] :
( ( countable @ A10 )
=> ( countable @ ( list @ A10 ) ) ) ).
thf(tcon_List_Olist___Nat_Osize_247,axiom,
! [A10: $tType] : ( size @ ( list @ A10 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_248,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_249,axiom,
condit1219197933456340205attice @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_250,axiom,
semiri1453513574482234551roduct @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_251,axiom,
ordere1937475149494474687imp_le @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinear__continuum__topology,axiom,
topolo8458572112393995274pology @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra,axiom,
real_V8999393235501362500lgebra @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom,
real_V2822296259951069270ebra_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_252,axiom,
semiri6575147826004484403cancel @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra,axiom,
real_V4412858255891104859lgebra @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oordered__real__vector,axiom,
real_V5355595471888546746vector @ real ).
thf(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_253,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_254,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_255,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_256,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_257,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_258,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_259,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_260,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_261,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_262,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_263,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_264,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_265,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_266,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_267,axiom,
linord181362715937106298miring @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
real_V2191834092415804123ebra_1 @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
real_V8037385150606011577_space @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_268,axiom,
topolo4211221413907600880p_mult @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
topolo7287701948861334536_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
topolo8386298272705272623_space @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_269,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_270,axiom,
semiri3467727345109120633visors @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
real_V6157519004096292374lgebra @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
real_V7819770556892013058_space @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_271,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_272,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_273,axiom,
archim2362893244070406136eiling @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
real_V4867850818363320053vector @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add_274,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_275,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_276,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_277,axiom,
ring_15535105094025558882visors @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__field,axiom,
real_V7773925162809079976_field @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__monoid__add_278,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_279,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_280,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_281,axiom,
comm_s4317794764714335236cancel @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_282,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_283,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_284,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_285,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_286,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_287,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_288,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_289,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_290,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_291,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_292,axiom,
semilattice_inf @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_293,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_294,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_295,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_296,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_297,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_298,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_299,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_300,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_301,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_302,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_303,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_304,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_305,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_306,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_307,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_308,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_309,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_310,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_311,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_312,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_313,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_314,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_315,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_316,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_317,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_318,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_319,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_320,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_321,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide_322,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_323,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_324,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_325,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_326,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_327,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_328,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_329,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_330,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_331,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_332,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_333,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_334,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_335,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_336,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_337,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom_338,axiom,
semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_339,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_340,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_341,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_342,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_343,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_344,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_345,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_346,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Rings_Oring_347,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_348,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_349,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_350,axiom,
dvd @ real ).
thf(tcon_String_Ochar___Countable_Ocountable_351,axiom,
countable @ char ).
thf(tcon_String_Ochar___Nat_Osize_352,axiom,
size @ char ).
thf(tcon_Sum__Type_Osum___Countable_Ocountable_353,axiom,
! [A10: $tType,A19: $tType] :
( ( ( countable @ A10 )
& ( countable @ A19 ) )
=> ( countable @ ( sum_sum @ A10 @ A19 ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_354,axiom,
! [A10: $tType,A19: $tType] : ( size @ ( sum_sum @ A10 @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_355,axiom,
! [A10: $tType] : ( condit1219197933456340205attice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_356,axiom,
! [A10: $tType] : ( counta3822494911875563373attice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_357,axiom,
! [A10: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_358,axiom,
! [A10: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_359,axiom,
! [A10: $tType] : ( comple6319245703460814977attice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_360,axiom,
! [A10: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_361,axiom,
! [A10: $tType] : ( semilattice_sup @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_362,axiom,
! [A10: $tType] : ( semilattice_inf @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_363,axiom,
! [A10: $tType] : ( bounded_lattice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_364,axiom,
! [A10: $tType] : ( order_top @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_365,axiom,
! [A10: $tType] : ( order_bot @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_366,axiom,
! [A10: $tType] : ( preorder @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_367,axiom,
! [A10: $tType] : ( lattice @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_368,axiom,
! [A10: $tType] : ( order @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_369,axiom,
! [A10: $tType] : ( top @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_370,axiom,
! [A10: $tType] : ( ord @ ( filter @ A10 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_371,axiom,
! [A10: $tType] : ( bot @ ( filter @ A10 ) ) ).
thf(tcon_Option_Ooption___Countable_Ocountable_372,axiom,
! [A10: $tType] :
( ( countable @ A10 )
=> ( countable @ ( option @ A10 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_373,axiom,
! [A10: $tType] : ( size @ ( option @ A10 ) ) ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_374,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_375,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_376,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_377,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_378,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_379,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_380,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_381,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_382,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_383,axiom,
real_V768167426530841204y_dist @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_384,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_385,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_386,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_387,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_388,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_389,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_390,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_391,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_392,axiom,
real_V6157519004096292374lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_393,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_394,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_395,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_396,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_397,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_398,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_399,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_400,axiom,
comm_s4317794764714335236cancel @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_401,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_402,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_403,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_404,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_405,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_406,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_407,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_408,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_409,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_410,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_411,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_412,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_413,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_414,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_415,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_416,axiom,
comm_semiring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_417,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_418,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_419,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_420,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_421,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_422,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_423,axiom,
idom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_424,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_425,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_426,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_427,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_428,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_429,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_430,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_431,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_432,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_433,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_434,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom_435,axiom,
semidom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_436,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_437,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_438,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_439,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_440,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_441,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_442,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_443,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_444,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_445,axiom,
dvd @ complex ).
thf(tcon_Typerep_Otyperep___Countable_Ocountable_446,axiom,
countable @ typerep ).
thf(tcon_Typerep_Otyperep___Nat_Osize_447,axiom,
size @ typerep ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_448,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_449,axiom,
counta4013691401010221786attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_450,axiom,
condit1219197933456340205attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_451,axiom,
counta3822494911875563373attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_452,axiom,
comple592849572758109894attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_453,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_454,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_455,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_456,axiom,
bounde4967611905675639751up_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_457,axiom,
bounde4346867609351753570nf_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_458,axiom,
comple6319245703460814977attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_459,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_460,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_461,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_462,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_463,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_464,axiom,
comple9053668089753744459l_ccpo @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_465,axiom,
semilattice_sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_466,axiom,
semilattice_inf @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_467,axiom,
bounded_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_468,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_469,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_470,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_471,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_472,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_473,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_474,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_475,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_476,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_477,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_478,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_479,axiom,
comm_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_480,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_481,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_482,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_483,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable_Ocountable_484,axiom,
countable @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_485,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_486,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_487,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_488,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_489,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_490,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_491,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_492,axiom,
lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_493,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_494,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_495,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Otop_496,axiom,
top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_497,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Obot_498,axiom,
bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_499,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_500,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_501,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_502,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_503,axiom,
dvd @ extended_enat ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_504,axiom,
! [A10: $tType,A19: $tType] :
( ( ( topolo4958980785337419405_space @ A10 )
& ( topolo4958980785337419405_space @ A19 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A10 @ A19 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_505,axiom,
! [A10: $tType,A19: $tType] :
( ( ( topological_t2_space @ A10 )
& ( topological_t2_space @ A19 ) )
=> ( topological_t2_space @ ( product_prod @ A10 @ A19 ) ) ) ).
thf(tcon_Product__Type_Oprod___Countable_Ocountable_506,axiom,
! [A10: $tType,A19: $tType] :
( ( ( countable @ A10 )
& ( countable @ A19 ) )
=> ( countable @ ( product_prod @ A10 @ A19 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_507,axiom,
! [A10: $tType,A19: $tType] : ( size @ ( product_prod @ A10 @ A19 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_508,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_509,axiom,
counta4013691401010221786attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_510,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_511,axiom,
counta3822494911875563373attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_512,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_513,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_514,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_515,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_516,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_517,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_518,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_519,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_520,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_521,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_522,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_523,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_524,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable_Ocountable_525,axiom,
countable @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_526,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_527,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_528,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_529,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_530,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_531,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_532,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_533,axiom,
uminus @ product_unit ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_534,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_535,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_536,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_537,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_538,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_539,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_540,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_541,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_542,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_543,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_544,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_545,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_546,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_547,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_548,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_549,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_550,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_551,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_552,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_553,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_554,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_555,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_556,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_557,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_558,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_559,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_560,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_561,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_562,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_563,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_564,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_565,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_566,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_567,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_568,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_569,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_570,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_571,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_572,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_573,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_574,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_575,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_576,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_577,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_578,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_579,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_580,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_581,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_582,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_583,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_584,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_585,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_586,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_587,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_588,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_589,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_590,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_591,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_592,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_593,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_594,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_595,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_596,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_597,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_598,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_599,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_600,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_601,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_602,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_603,axiom,
idom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_604,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_605,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_606,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_607,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_608,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_609,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_610,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_611,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_612,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_613,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_614,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_615,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_616,axiom,
semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_617,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_618,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_619,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_620,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_621,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_622,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_623,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_624,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_625,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_626,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_627,axiom,
dvd @ code_integer ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_628,axiom,
bit_un5681908812861735899ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_629,axiom,
euclid5411537665997757685th_nat @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_630,axiom,
ordere1937475149494474687imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_631,axiom,
euclid3128863361964157862miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_632,axiom,
euclid4440199948858584721cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_633,axiom,
semiri6575147826004484403cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_634,axiom,
strict9044650504122735259up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_635,axiom,
ordere580206878836729694up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_636,axiom,
ordere2412721322843649153imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_637,axiom,
bit_se359711467146920520ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_638,axiom,
linord2810124833399127020strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_639,axiom,
strict7427464778891057005id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_640,axiom,
ordere8940638589300402666id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_641,axiom,
euclid3725896446679973847miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_642,axiom,
semiri2026040879449505780visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_643,axiom,
linord181362715937106298miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_644,axiom,
linord8928482502909563296strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_645,axiom,
semiri3467727345109120633visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_646,axiom,
ordere6658533253407199908up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_647,axiom,
ordere6911136660526730532id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_648,axiom,
cancel2418104881723323429up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_649,axiom,
cancel1802427076303600483id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_650,axiom,
comm_s4317794764714335236cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_651,axiom,
bit_semiring_bits @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_652,axiom,
ordere2520102378445227354miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_653,axiom,
cancel_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_654,axiom,
linordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_655,axiom,
ordered_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_656,axiom,
linordered_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_657,axiom,
ab_semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_658,axiom,
semiring_1_cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_659,axiom,
algebraic_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_660,axiom,
comm_monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_661,axiom,
comm_monoid_diff @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_662,axiom,
ab_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_663,axiom,
ordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_664,axiom,
semiring_parity @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_665,axiom,
comm_monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_666,axiom,
semiring_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_667,axiom,
comm_semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_668,axiom,
comm_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_669,axiom,
semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_670,axiom,
semidom_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_671,axiom,
semidom_divide @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_672,axiom,
semiring_numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_673,axiom,
semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_674,axiom,
zero_less_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_675,axiom,
comm_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_676,axiom,
semiring_char_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_677,axiom,
zero_neq_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_678,axiom,
preorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_679,axiom,
linorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_680,axiom,
monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_681,axiom,
monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_682,axiom,
semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_683,axiom,
semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_684,axiom,
mult_zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_685,axiom,
order @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_686,axiom,
semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom_687,axiom,
semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_688,axiom,
ord @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Power_Opower_689,axiom,
power @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Onumeral_690,axiom,
numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_691,axiom,
zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_692,axiom,
one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Odvd_693,axiom,
dvd @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osize_694,axiom,
size @ code_natural ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_695,axiom,
size @ vEBT_VEBT ).
thf(tcon_Record_Otuple__isomorphism___Nat_Osize_696,axiom,
! [A10: $tType,A19: $tType,A20: $tType] : ( size @ ( tuple_isomorphism @ A10 @ A19 @ A20 ) ) ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P2: A > $o] :
( ( P2 @ ( fChoice @ A @ P2 ) )
= ( ? [X7: A] : ( P2 @ X7 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( info
= ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) ) ).
%------------------------------------------------------------------------------