TPTP Problem File: ITP281^4.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP281^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Intf_Functional 00357_014232
% 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 : 0077_VEBT_Intf_Functional_00357_014232 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 9520 (2991 unt; 553 typ; 0 def)
% Number of atoms : 29108 (10080 equ; 0 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 190946 (2441 ~; 328 |;2250 &;172366 @)
% ( 0 <=>;13561 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 7 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 3781 (3781 >; 0 *; 0 +; 0 <<)
% Number of symbols : 546 ( 543 usr; 8 con; 0-8 aty)
% Number of variables : 29840 (2854 ^;25744 !; 824 ?;29840 :)
% ( 418 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-18 17:39:43.185
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_t_VEBT__Definitions_OVEBT,type,
vEBT_VEBT: $tType ).
thf(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Complex_Ocomplex,type,
complex: $tType ).
thf(ty_t_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 ).
% Explicit typings (538)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Oring__gcd,type,
ring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom,type,
semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__divide,type,
idom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__modulo,type,
idom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Transcendental_Oln,type,
ln:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Oring__parity,type,
ring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__lattice,type,
finite_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_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_Complete__Lattices_OInf,type,
complete_Inf:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[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_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_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot1__space,type,
topological_t1_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot2__space,type,
topological_t2_space:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__group__add,type,
topolo1633459387980952147up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__monoid__add,type,
topolo6943815403480290642id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__field,type,
real_V7773925162809079976_field:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__monoid__mult,type,
topolo1898628316856586783d_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
real_V4867850818363320053vector:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
semiri6843258321239162965malize:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__ab__group__add,type,
topolo1287966508704411220up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V7819770556892013058_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo8386298272705272623_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo7287701948861334536_space:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring,type,
euclid5891614535332579305n_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__semigroup__mult,type,
topolo4211221413907600880p_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
real_V8037385150606011577_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__algebra__1,type,
real_V2191834092415804123ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
semiri2026040879449505780visors:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Oorder__topology,type,
topolo2564578578187576103pology:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__comm__monoid__add,type,
topolo5987344860129210374id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ounbounded__dense__linorder,type,
unboun7993243217541854897norder:
!>[A: $tType] : $o ).
thf(sy_cl_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_Limits_Otopological__comm__monoid__mult,type,
topolo4987421752381908075d_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type,
real_V5047593784448816457lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__field,type,
real_V3459762299906320749_field:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Odiscrete__topology,type,
topolo8865339358273720382pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
topolo1944317154257567458pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo4958980785337419405_space:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V822414075346904944vector:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oordered__real__vector,type,
real_V5355595471888546746vector:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra,type,
real_V4412858255891104859lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1,type,
real_V2822296259951069270ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__boolean__algebra,type,
comple489889107523837845lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra,type,
real_V8999393235501362500lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
topolo3112930676232923870pology:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid4440199948858584721cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
euclid3128863361964157862miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinear__continuum__topology,type,
topolo8458572112393995274pology:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit5016429287641298734tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
euclid8789492081693882211th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice,type,
counta3822494911875563373attice:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_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_Basic__BNF__LFPs_Oprod_Osize__prod,type,
basic_BNF_size_prod:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
bit_or3848514188828904588eg_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
thf(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onegative,type,
code_negative: num > code_integer ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Code__Numeral_Opositive,type,
code_positive: num > code_integer ).
thf(sy_c_Code__Target__Int_Onegative,type,
code_Target_negative: num > int ).
thf(sy_c_Code__Target__Int_Opositive,type,
code_Target_positive: num > int ).
thf(sy_c_Code__Target__Nat_Oint__of__nat,type,
code_T6385005292777649522of_nat: nat > int ).
thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_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_Deriv_Ohas__derivative,type,
has_derivative:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__field__derivative,type,
has_field_derivative:
!>[A: $tType] : ( ( A > A ) > A > ( filter @ A ) > $o ) ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_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_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_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_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_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_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_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > ( product_prod @ int @ int ) ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
groups1027152243600224163dd_sum:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_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_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMax,type,
lattices_Max:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMin,type,
lattices_Min:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Limits_OBfun,type,
bfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Limits_Oat__infinity,type,
at_infinity:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_List_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_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_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_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_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_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__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : ( ( list @ ( set @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__project,type,
map_project:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_List_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_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[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_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osplice__rel,type,
splice_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > int > ( list @ int ) ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > ( list @ int ) > ( list @ int ) ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
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_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_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_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : 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_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_Otop__class_Otop,type,
top_top:
!>[A: $tType] : 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_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Rat_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Real__Vector__Spaces_OReals,type,
real_Vector_Reals:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear,type,
real_V3181309239436604168linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V557655796197034286t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
real_V7770717601297561774m_norm:
!>[A: $tType] : ( A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oof__real,type,
real_Vector_of_real:
!>[A: $tType] : ( real > A ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V8093663219630862766scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Series_Osuminf,type,
suminf:
!>[A: $tType] : ( ( nat > A ) > A ) ).
thf(sy_c_Series_Osummable,type,
summable:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Series_Osums,type,
sums:
!>[A: $tType] : ( ( nat > A ) > A > $o ) ).
thf(sy_c_Set_OBex,type,
bex:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow2:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Ofilter,type,
filter3:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > 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_Oascii__of,type,
ascii_of: char > char ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_String_Ochar__of__integer,type,
char_of_integer: code_integer > char ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
thf(sy_c_String_Ointeger__of__char,type,
integer_of_char: char > code_integer ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
thf(sy_c_Topological__Spaces_Ocontinuous,type,
topolo3448309680560233919inuous:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Topological__Spaces_Omonoseq,type,
topological_monoseq:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
topolo1002775350975398744n_open:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_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_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_Transcendental_Oarccos,type,
arccos: real > real ).
thf(sy_c_Transcendental_Oarcosh,type,
arcosh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oarcsin,type,
arcsin: real > real ).
thf(sy_c_Transcendental_Oarctan,type,
arctan: real > real ).
thf(sy_c_Transcendental_Oarsinh,type,
arsinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oartanh,type,
artanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocos,type,
cos:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Ocosh,type,
cosh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocot,type,
cot:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Odiffs,type,
diffs:
!>[A: $tType] : ( ( nat > A ) > nat > A ) ).
thf(sy_c_Transcendental_Oexp,type,
exp:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oln__class_Oln,type,
ln_ln:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Opowr,type,
powr:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Transcendental_Osin,type,
sin:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Osinh,type,
sinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otan,type,
tan:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otanh,type,
tanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
vEBT_T_i_n_s_e_r_t: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
vEBT_T5076183648494686801_t_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
vEBT_T9217963907923527482_t_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
vEBT_T_m_a_x_t: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
vEBT_T_m_e_m_b_e_r: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
vEBT_T8099345112685741742_r_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
vEBT_T5837161174952499735_r_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
vEBT_T_m_i_n_t: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
vEBT_T_m_i_n_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
vEBT_T_p_r_e_d: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
vEBT_T_p_r_e_d2: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
vEBT_T_p_r_e_d_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
vEBT_T_p_r_e_d_rel2: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
vEBT_T_s_u_c_c: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
vEBT_T_s_u_c_c2: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
vEBT_T_s_u_c_c_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
vEBT_T_s_u_c_c_rel2: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
vEBT_Node: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_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_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__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
vEBT_T_d_e_l_e_t_e: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
vEBT_T8441311223069195367_e_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
vEBT_V1232361888498592333_e_t_e: vEBT_VEBT > nat > nat ).
thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
vEBT_V6368547301243506412_e_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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__Height_OVEBT__internal_Oheight,type,
vEBT_VEBT_height: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $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__Intf__Functional_Oexperiment7207664_Otest,type,
vEBT_Intf_test: nat > ( list @ nat ) > ( list @ nat ) > ( list @ nat ) ).
thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_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_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_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__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
vEBT_V8646137997579335489_i_l_d: nat > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
vEBT_V8346862874174094_d_u_p: nat > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
vEBT_V1247956027447740395_p_rel: nat > nat > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
vEBT_V5144397997797733112_d_rel: nat > nat > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
vEBT_VEBT_cnt: vEBT_VEBT > real ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
vEBT_VEBT_cnt2: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
vEBT_VEBT_cnt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
vEBT_VEBT_cnt_rel2: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
vEBT_VEBT_space: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
vEBT_VEBT_space2: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
vEBT_VEBT_space_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
vEBT_VEBT_space_rel2: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
vEBT_is_succ_in_set: ( set @ nat ) > nat > nat > $o ).
thf(sy_c_VEBT__Succ_Ovebt__succ,type,
vEBT_vebt_succ: vEBT_VEBT > nat > ( option @ nat ) ).
thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
vEBT_vebt_succ_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
thf(sy_c_Wellfounded_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_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_xs,type,
xs: list @ nat ).
thf(sy_v_ys,type,
ys: list @ nat ).
% Relevant facts (8181)
thf(fact_0_zero__less__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_power2
thf(fact_1_power2__less__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% power2_less_eq_zero_iff
thf(fact_2_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_3_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_4_zero__eq__power2,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A] :
( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_power2
thf(fact_5_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N: nat] :
( ( ( power_power @ A @ A2 @ N )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% power_eq_0_iff
thf(fact_6_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_7_power__zero__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [K: num] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K ) )
= ( zero_zero @ A ) ) ) ).
% power_zero_numeral
thf(fact_8_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_9_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_10_power2__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% power2_less_0
thf(fact_11_zero__le__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% zero_le_power2
thf(fact_12_power__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_not_zero
thf(fact_13_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_le_power
thf(fact_14_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono
thf(fact_15_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_less_power
thf(fact_16_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_17_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_18_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_19_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_20_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_21_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B2 @ N ) )
= ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_22_less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% less_exp
thf(fact_23_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_24_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_25_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ M @ ( power_power @ nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_26_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_27_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_28_map__eq__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,G: B > A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ B @ A @ G @ Xs ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ ( set2 @ B @ Xs ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_29_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_30_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_31_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_32_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_33_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_34_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_35_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_36_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_37_semiring__norm_I83_J,axiom,
! [N: num] :
( one2
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_38_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one2 ) ).
% semiring_norm(85)
thf(fact_39_numeral__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N: num] :
( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ M @ N ) ) ) ).
% numeral_less_iff
thf(fact_40_numeral__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( numeral_numeral @ A @ M )
= ( numeral_numeral @ A @ N ) )
= ( M = N ) ) ) ).
% numeral_eq_iff
thf(fact_41_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_42_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X2: A] : X2 )
= ( ^ [Xs2: list @ A] : Xs2 ) ) ).
% map_ident
thf(fact_43_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X3: A] :
( ( F2 @ X3 )
= ( G @ X3 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_48_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_49_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% semiring_norm(68)
thf(fact_50_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_51_numeral__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ) ).
% numeral_le_iff
thf(fact_52_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y2: A] :
( ( ord_less @ B @ ( F2 @ Y2 ) @ ( F2 @ X3 ) )
=> ( P @ Y2 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct
thf(fact_53_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y2: A] :
( ( ord_less @ B @ ( F2 @ Y2 ) @ ( F2 @ X3 ) )
=> ( P @ Y2 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_54_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq @ num @ X @ one2 )
= ( X = one2 ) ) ).
% le_num_One_iff
thf(fact_55_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_56_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_57_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_58_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_59_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_60_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_61_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_62_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_63_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y2: A] :
( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X3 ) )
& ~ ( P @ Y2 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_64_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_65_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_66_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_67_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_68_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_69_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_70_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B3 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X2 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_71_list_Omap__ident,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A
@ ^ [X2: A] : X2
@ T2 )
= T2 ) ).
% list.map_ident
thf(fact_72_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_73_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_74_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_75_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_76_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_77_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_78_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_79_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_80_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_81_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_82_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X3: A] :
( ( ( V @ X3 )
= ( zero_zero @ nat ) )
=> ( P @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
=> ( ~ ( P @ X3 )
=> ? [Y2: A] :
( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X3 ) )
& ~ ( P @ Y2 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_83_realpow__pos__nth,axiom,
! [N: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
& ( ( power_power @ real @ R @ N )
= A2 ) ) ) ) ).
% realpow_pos_nth
thf(fact_84_realpow__pos__nth__unique,axiom,
! [N: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ( power_power @ real @ X3 @ N )
= A2 )
& ! [Y2: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
& ( ( power_power @ real @ Y2 @ N )
= A2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_85_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_86_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_87_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_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_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_90_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_91_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M3: nat,N3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_92_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less @ nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_93_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( M != N )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_94_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_95_list_Omap__cong,axiom,
! [B: $tType,A: $tType,X: list @ A,Ya: list @ A,F2: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( set2 @ A @ Ya ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_96_list_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: list @ A,F2: A > B,G: A > B] :
( ! [Z: A] :
( ( member @ A @ Z @ ( set2 @ A @ X ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_97_list_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: list @ A,Xa: list @ A,F2: A > B,Fa: A > B] :
( ! [Z: A,Za: A] :
( ( member @ A @ Z @ ( set2 @ A @ X ) )
=> ( ( member @ A @ Za @ ( set2 @ A @ Xa ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map @ A @ B @ F2 @ X )
= ( map @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_98_map__ext,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B,G: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ G @ Xs ) ) ) ).
% map_ext
thf(fact_99_map__idI,axiom,
! [A: $tType,Xs: list @ A,F2: A > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( F2 @ X3 )
= X3 ) )
=> ( ( map @ A @ A @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_100_map__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F2: A > B,G: A > B] :
( ( Xs = Ys )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map @ A @ B @ F2 @ Xs )
= ( map @ A @ B @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_101_ex__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ B,F2: A > B] :
( ( ? [Xs2: list @ A] :
( Ys
= ( map @ A @ B @ F2 @ Xs2 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ ( set2 @ B @ Ys ) )
=> ? [Y4: A] :
( X2
= ( F2 @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_102_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_103_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_104_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_105_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_106_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_107_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_108_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_109_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_110_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_111_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_112_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_113_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_114_power__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,L: num] :
( ( power_power @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ nat @ L ) )
= ( numeral_numeral @ A @ ( pow @ K @ L ) ) ) ) ).
% power_numeral
thf(fact_115_insertCI,axiom,
! [A: $tType,A2: A,B3: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_116_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_117_insert__absorb2,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A3 ) )
= ( insert @ A @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_118_insert__subset,axiom,
! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B3 )
= ( ( member @ A @ X @ B3 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_119_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_120_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_121_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_122_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_123_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_124_subsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% subsetI
thf(fact_125_psubsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% psubsetI
thf(fact_126_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_127_i0__less,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( N
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_128_in__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B3 ) ) ) ).
% in_mono
thf(fact_129_subsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_130_psubsetE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% psubsetE
thf(fact_131_equalityE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_132_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_133_equalityD1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_134_equalityD2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_135_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_136_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A4 )
=> ( member @ A @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_137_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_138_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_139_subset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_140_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2 )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_141_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_142_Collect__subset,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_143_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ ^ [X2: A] : ( member @ A @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_144_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_less_eq @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= ( N
= ( zero_zero @ extended_enat ) ) ) ).
% ile0_eq
thf(fact_145_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_146_psubset__imp__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_147_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_less @ extended_enat @ N @ ( zero_zero @ extended_enat ) ) ).
% not_iless0
thf(fact_148_psubset__subset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_149_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
& ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_150_subset__psubset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ B3 @ C3 )
=> ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_151_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_152_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N2: extended_enat] :
( ! [M2: extended_enat] :
( ( ord_less @ extended_enat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_153_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one2 )
= X ) ).
% pow.simps(1)
thf(fact_154_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_155_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_156_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_157_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F2: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_158_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_159_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_160_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
| ~ ( ord_less_eq @ A @ A2 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_161_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_162_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_163_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_164_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_165_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_166_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_167_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_168_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_169_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% antisym
thf(fact_170_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_171_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_172_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_173_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_174_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_175_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_176_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_177_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_178_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_179_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_180_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_181_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( ord_less_eq @ A @ A2 @ B2 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( B2 != A2 ) ) ) ) ).
% nle_le
thf(fact_182_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(2)
thf(fact_183_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_184_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_185_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_186_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_187_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_188_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_189_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_190_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_191_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_192_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F2: A > B,C2: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_193_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_194_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_195_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order_less_asym'
thf(fact_196_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_197_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_198_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_199_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_200_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_201_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_202_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_203_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_204_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B6: A] :
( ( ord_less @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_205_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X4: A] : ( P2 @ X4 ) )
= ( ^ [P3: A > $o] :
? [N3: A] :
( ( P3 @ N3 )
& ! [M3: A] :
( ( ord_less @ A @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_206_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_207_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_208_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_209_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_210_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y2: A] :
( ( ord_less @ A @ Y2 @ X3 )
=> ( P @ Y2 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_211_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_212_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_213_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_214_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_215_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_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_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_218_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_219_subset__insertI2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% subset_insertI2
thf(fact_220_subset__insertI,axiom,
! [A: $tType,B3: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ B3 ) ) ).
% subset_insertI
thf(fact_221_subset__insert,axiom,
! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_222_insert__mono,axiom,
! [A: $tType,C3: set @ A,D: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ D )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C3 ) @ ( insert @ A @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_223_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ? [B7: set @ A] :
( ( A3
= ( insert @ A @ A2 @ B7 ) )
& ~ ( member @ A @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_224_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A3: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A3 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_225_insert__eq__iff,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A,B3: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ B2 @ B3 )
=> ( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B3 ) )
& ( ( A2 != B2 )
=> ? [C4: set @ A] :
( ( A3
= ( insert @ A @ B2 @ C4 ) )
& ~ ( member @ A @ B2 @ C4 )
& ( B3
= ( insert @ A @ A2 @ C4 ) )
& ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_226_insert__absorb,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_227_insert__ident,axiom,
! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ~ ( member @ A @ X @ B3 )
=> ( ( ( insert @ A @ X @ A3 )
= ( insert @ A @ X @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_228_Set_Oset__insert,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ~ ! [B7: set @ A] :
( ( A3
= ( insert @ A @ X @ B7 ) )
=> ( member @ A @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_229_insertI2,axiom,
! [A: $tType,A2: A,B3: set @ A,B2: A] :
( ( member @ A @ A2 @ B3 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertI2
thf(fact_230_insertI1,axiom,
! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% insertI1
thf(fact_231_insertE,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_232_insert__Collect,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( insert @ A @ A2 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U: A] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_233_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A5: A,B4: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( X2 = A5 )
| ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_234_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_235_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_236_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_237_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_238_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_239_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_240_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_241_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_242_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_243_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_244_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_245_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_246_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_247_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order_neq_le_trans
thf(fact_248_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order_le_neq_trans
thf(fact_249_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_250_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_251_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_252_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ) ).
% order_less_le
thf(fact_253_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ) ).
% order_le_less
thf(fact_254_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_255_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_256_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ~ ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_257_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_258_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_259_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_260_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less @ A @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_261_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_262_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_263_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ~ ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_264_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_265_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_266_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_267_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less @ A @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_268_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_269_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ~ ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_270_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_271_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_272_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_273_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_274_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( ord_less @ A @ A2 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% nless_le
thf(fact_275_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_276_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_277_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B8: B,A7: B] :
( ( ~ ( ord_less_eq @ B @ B8 @ A7 ) )
= ( ord_less @ B @ A7 @ B8 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_278_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one2
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_279_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_280_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_281_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_282_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_283_insert__subsetI,axiom,
! [A: $tType,X: A,A3: set @ A,X5: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_284_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F2 @ Y3 ) @ B2 ) )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ ( F2 @ Y2 ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_285_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_286_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_287_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ int ) ) ).
% not_exp_less_eq_0_int
thf(fact_288_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C5: A] :
( ( ord_less_eq @ A @ A2 @ C5 )
& ( ord_less_eq @ A @ C5 @ B2 )
& ! [X6: A] :
( ( ( ord_less_eq @ A @ A2 @ X6 )
& ( ord_less @ A @ X6 @ C5 ) )
=> ( P @ X6 ) )
& ! [D2: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less @ A @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D2 @ C5 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_289_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ~ ( ord_less_eq @ A @ X6 @ T2 ) ) ) ).
% pinf(6)
thf(fact_290_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ( ord_less_eq @ A @ T2 @ X6 ) ) ) ).
% pinf(8)
thf(fact_291_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( semiring_1_of_nat @ A @ N ) )
= ( M = N ) ) ) ).
% of_nat_eq_iff
thf(fact_292_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_293_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_294_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_295_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_296_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_297_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_iff
thf(fact_298_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% of_nat_le_iff
thf(fact_299_of__nat__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% of_nat_power
thf(fact_300_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [B2: nat,W2: nat,X: nat] :
( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 )
= ( semiring_1_of_nat @ A @ X ) )
= ( ( power_power @ nat @ B2 @ W2 )
= X ) ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_301_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X: nat,B2: nat,W2: nat] :
( ( ( semiring_1_of_nat @ A @ X )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( X
= ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_302_of__nat__le__0__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_le_0_iff
thf(fact_303_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_304_numeral__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X: num,N: nat,Y: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
= ( semiring_1_of_nat @ A @ Y ) )
= ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
= Y ) ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_305_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [Y: nat,X: num,N: nat] :
( ( ( semiring_1_of_nat @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( Y
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_306_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_307_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_308_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_309_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_310_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_311_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A2 @ N ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% even_power
thf(fact_312_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_313_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_314_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_315_even__of__nat,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ N ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% even_of_nat
thf(fact_316_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( ( numeral_numeral @ nat @ W2 )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_317_psubsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% psubsetD
thf(fact_318_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ ^ [X2: A] : ( member @ A @ X2 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_319_psubset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ B3 @ C3 )
=> ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% psubset_trans
thf(fact_320_verit__la__generic,axiom,
! [A2: int,X: int] :
( ( ord_less_eq @ int @ A2 @ X )
| ( A2 = X )
| ( ord_less_eq @ int @ X @ A2 ) ) ).
% verit_la_generic
thf(fact_321_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_322_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% less_fun_def
thf(fact_323_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_324_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_325_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_326_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_327_dvd__power__same,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) ) ) ) ).
% dvd_power_same
thf(fact_328_int__ops_I3_J,axiom,
! [N: num] :
( ( semiring_1_of_nat @ int @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ int @ N ) ) ).
% int_ops(3)
thf(fact_329_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_330_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_331_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_332_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_less_as_int
thf(fact_333_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_334_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_335_of__nat__less__0__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat] :
~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% of_nat_less_0_iff
thf(fact_336_even__numeral,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: num] : ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ N ) ) ) ) ).
% even_numeral
thf(fact_337_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_338_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_imp_less
thf(fact_339_of__nat__mono,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% of_nat_mono
thf(fact_340_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N: nat,M: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_341_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat,B2: A,M: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ B2 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ B2 ) ) ) ) ).
% power_le_dvd
thf(fact_342_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% le_imp_power_dvd
thf(fact_343_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_344_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_345_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_346_minf_I11_J,axiom,
! [C: $tType,D3: $tType] :
( ( ord @ C )
=> ! [F4: D3] :
? [Z: C] :
! [X6: C] :
( ( ord_less @ C @ X6 @ Z )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_347_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ~ ( ord_less @ A @ T2 @ X6 ) ) ) ).
% minf(7)
thf(fact_348_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ( ord_less @ A @ X6 @ T2 ) ) ) ).
% minf(5)
thf(fact_349_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ( X6 != T2 ) ) ) ).
% minf(4)
thf(fact_350_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ( X6 != T2 ) ) ) ).
% minf(3)
thf(fact_351_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_352_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_353_pinf_I11_J,axiom,
! [C: $tType,D3: $tType] :
( ( ord @ C )
=> ! [F4: D3] :
? [Z: C] :
! [X6: C] :
( ( ord_less @ C @ Z @ X6 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_354_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ( ord_less @ A @ T2 @ X6 ) ) ) ).
% pinf(7)
thf(fact_355_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ~ ( ord_less @ A @ X6 @ T2 ) ) ) ).
% pinf(5)
thf(fact_356_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ( X6 != T2 ) ) ) ).
% pinf(4)
thf(fact_357_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ( X6 != T2 ) ) ) ).
% pinf(3)
thf(fact_358_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_359_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_360_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A2: A] :
? [B6: A] :
( ( ord_less @ A @ A2 @ B6 )
| ( ord_less @ A @ B6 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_361_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y2 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_362_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_odd
thf(fact_363_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_364_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_365_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% zero_le_odd_power
thf(fact_366_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_le_even_power
thf(fact_367_Collect__restrict,axiom,
! [A: $tType,X5: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X5 )
& ( P @ X2 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_368_prop__restrict,axiom,
! [A: $tType,X: A,Z5: set @ A,X5: set @ A,P: A > $o] :
( ( member @ A @ X @ Z5 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z5
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X5 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_369_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_370_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_371_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_372_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ~ ( ord_less_eq @ A @ T2 @ X6 ) ) ) ).
% minf(8)
thf(fact_373_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ( ord_less_eq @ A @ X6 @ T2 ) ) ) ).
% minf(6)
thf(fact_374_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_375_numeral__less__real__of__nat__iff,axiom,
! [W2: num,N: nat] :
( ( ord_less @ real @ ( numeral_numeral @ real @ W2 ) @ ( semiring_1_of_nat @ real @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ W2 ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_376_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W2: num] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( numeral_numeral @ real @ W2 ) )
= ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ W2 ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_377_of__nat__less__two__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat] : ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% of_nat_less_two_power
thf(fact_378_int__eq__iff__numeral,axiom,
! [M: nat,V2: num] :
( ( ( semiring_1_of_nat @ int @ M )
= ( numeral_numeral @ int @ V2 ) )
= ( M
= ( numeral_numeral @ nat @ V2 ) ) ) ).
% int_eq_iff_numeral
thf(fact_379_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N ) @ ( semiring_1_of_nat @ real @ M ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_380_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_381_dvd__0__left__iff,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left_iff
thf(fact_382_of__nat__dvd__iff,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( dvd_dvd @ nat @ M @ N ) ) ) ).
% of_nat_dvd_iff
thf(fact_383_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ M @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ).
% dvd_pos_nat
thf(fact_384_pos__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% pos_int_cases
thf(fact_385_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( dvd_dvd @ nat @ M @ N ) ) ).
% int_dvd_int_iff
thf(fact_386_complete__real,axiom,
! [S2: set @ real] :
( ? [X6: real] : ( member @ real @ X6 @ S2 )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member @ real @ X3 @ S2 )
=> ( ord_less_eq @ real @ X3 @ Z4 ) )
=> ? [Y3: real] :
( ! [X6: real] :
( ( member @ real @ X6 @ S2 )
=> ( ord_less_eq @ real @ X6 @ Y3 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ S2 )
=> ( ord_less_eq @ real @ X3 @ Z4 ) )
=> ( ord_less_eq @ real @ Y3 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_387_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
= ( ^ [A5: nat,B5: nat] :
( ( semiring_1_of_nat @ int @ A5 )
= ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_388_int__if,axiom,
! [P: $o,A2: nat,B2: nat] :
( ( P
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B2 ) )
= ( semiring_1_of_nat @ int @ A2 ) ) )
& ( ~ P
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B2 ) )
= ( semiring_1_of_nat @ int @ B2 ) ) ) ) ).
% int_if
thf(fact_389_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ int @ M )
= ( semiring_1_of_nat @ int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_390_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ord_less @ int @ M @ N )
=> ~ ( dvd_dvd @ int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_391_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( dvd_dvd @ int @ M @ N )
=> ( ( dvd_dvd @ int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_392_zdvd__imp__le,axiom,
! [Z3: int,N: int] :
( ( dvd_dvd @ int @ Z3 @ N )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ Z3 @ N ) ) ) ).
% zdvd_imp_le
thf(fact_393_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_394_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X2: real,Y4: real] :
( ( ord_less @ real @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_395_dvd__refl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ A2 ) ) ).
% dvd_refl
thf(fact_396_dvd__trans,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ C2 )
=> ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_trans
thf(fact_397_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_398_gcd__nat_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ~ ( ( dvd_dvd @ nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% gcd_nat.asym
thf(fact_399_gcd__nat_Orefl,axiom,
! [A2: nat] : ( dvd_dvd @ nat @ A2 @ A2 ) ).
% gcd_nat.refl
thf(fact_400_gcd__nat_Otrans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ).
% gcd_nat.trans
thf(fact_401_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
= ( ^ [A5: nat,B5: nat] :
( ( dvd_dvd @ nat @ A5 @ B5 )
& ( dvd_dvd @ nat @ B5 @ A5 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_402_gcd__nat_Oirrefl,axiom,
! [A2: nat] :
~ ( ( dvd_dvd @ nat @ A2 @ A2 )
& ( A2 != A2 ) ) ).
% gcd_nat.irrefl
thf(fact_403_gcd__nat_Oantisym,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% gcd_nat.antisym
thf(fact_404_gcd__nat_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_405_gcd__nat_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_406_gcd__nat_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_407_gcd__nat_Ostrict__iff__not,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
= ( ( dvd_dvd @ nat @ A2 @ B2 )
& ~ ( dvd_dvd @ nat @ B2 @ A2 ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_408_gcd__nat_Oorder__iff__strict,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A5: nat,B5: nat] :
( ( ( dvd_dvd @ nat @ A5 @ B5 )
& ( A5 != B5 ) )
| ( A5 = B5 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_409_gcd__nat_Ostrict__iff__order,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
= ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_410_gcd__nat_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( dvd_dvd @ nat @ A2 @ B2 ) ) ).
% gcd_nat.strict_implies_order
thf(fact_411_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( A2 != B2 ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_412_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_413_field__lbound__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D1: A,D22: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
=> ? [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
& ( ord_less @ A @ E @ D1 )
& ( ord_less @ A @ E @ D22 ) ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_414_dvd__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 )
=> ( A2
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left
thf(fact_415_gcd__nat_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_416_gcd__nat_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) )
& ( A2
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_417_gcd__nat_Oextremum__unique,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_418_gcd__nat_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
& ( ( zero_zero @ nat )
!= A2 ) ) ).
% gcd_nat.extremum_strict
thf(fact_419_gcd__nat_Oextremum,axiom,
! [A2: nat] : ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_420_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% zle_int
thf(fact_421_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N2: nat] :
( K
= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_422_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_423_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_424_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ A2 ) )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ B2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_425_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ A2 ) )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ B2 ) ) )
= ( ( dvd_dvd @ A @ A2 @ B2 )
& ~ ( dvd_dvd @ A @ B2 @ A2 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_426_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( K
= ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_427_even__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
| ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_unset_bit_iff
thf(fact_428_even__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
!= ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_flip_bit_iff
thf(fact_429_even__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( M
!= ( zero_zero @ nat ) ) ) ) ) ).
% even_set_bit_iff
thf(fact_430_pred__subset__eq,axiom,
! [A: $tType,R2: set @ A,S2: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R2 )
@ ^ [X2: A] : ( member @ A @ X2 @ S2 ) )
= ( ord_less_eq @ ( set @ A ) @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_431_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_432_nat__le__numeral__power__cancel__iff,axiom,
! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_433_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A2: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_434_nat__less__numeral__power__cancel__iff,axiom,
! [A2: int,X: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_435_log2__of__power__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_le
thf(fact_436_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_437_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_438_of__int__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W2: int,Z3: int] :
( ( ( ring_1_of_int @ A @ W2 )
= ( ring_1_of_int @ A @ Z3 ) )
= ( W2 = Z3 ) ) ) ).
% of_int_eq_iff
thf(fact_439_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_440_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_441_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_442_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% unset_bit_negative_int_iff
thf(fact_443_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% set_bit_negative_int_iff
thf(fact_444_flip__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% flip_bit_negative_int_iff
thf(fact_445_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
= N ) ).
% nat_int
thf(fact_446_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_447_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_448_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_449_of__int__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ A @ K ) ) ) ).
% of_int_numeral
thf(fact_450_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z3: int,N: num] :
( ( ( ring_1_of_int @ A @ Z3 )
= ( numeral_numeral @ A @ N ) )
= ( Z3
= ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_451_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_452_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z3: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% of_int_less_iff
thf(fact_453_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_numeral
thf(fact_454_of__int__of__nat__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_int_of_nat_eq
thf(fact_455_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: int,N: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z3 @ N ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z3 ) @ N ) ) ) ).
% of_int_power
thf(fact_456_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B2: int,W2: nat,X: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 )
= ( ring_1_of_int @ A @ X ) )
= ( ( power_power @ int @ B2 @ W2 )
= X ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_457_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: int,B2: int,W2: nat] :
( ( ( ring_1_of_int @ A @ X )
= ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( X
= ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_458_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z3 )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_459_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_460_zless__nat__conj,axiom,
! [W2: int,Z3: int] :
( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
& ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_461_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
= ( zero_zero @ int ) ) ) ) ).
% int_nat_eq
thf(fact_462_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_463_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_464_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_465_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_466_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_467_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_468_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int,N: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ int @ Z3 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_469_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z3: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N ) @ Z3 ) ) ) ).
% of_int_numeral_less_iff
thf(fact_470_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_471_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N: nat,Y: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
= Y ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_472_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_473_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_474_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_475_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_476_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_477_of__nat__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ A @ ( nat2 @ Z3 ) )
= ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_nat_nat
thf(fact_478_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( nat2 @ Y )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_479_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
= ( nat2 @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_480_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_481_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_482_unset__bit__nat__def,axiom,
( ( bit_se2638667681897837118et_bit @ nat )
= ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se2638667681897837118et_bit @ int @ M3 @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% unset_bit_nat_def
thf(fact_483_unset__bit__less__eq,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_484_set__bit__greater__eq,axiom,
! [K: int,N: nat] : ( ord_less_eq @ int @ K @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) ) ).
% set_bit_greater_eq
thf(fact_485_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_486_nat__numeral__as__int,axiom,
( ( numeral_numeral @ nat )
= ( ^ [I4: num] : ( nat2 @ ( numeral_numeral @ int @ I4 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_487_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_488_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
& ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_489_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
! [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_490_eq__nat__nat__iff,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z6 ) )
= ( Z3 = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_491_nat__mono__iff,axiom,
! [Z3: int,W2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_492_zless__nat__eq__int__zless,axiom,
! [M: nat,Z3: int] :
( ( ord_less @ nat @ M @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_493_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_494_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_495_int__eq__iff,axiom,
! [M: nat,Z3: int] :
( ( ( semiring_1_of_nat @ int @ M )
= Z3 )
= ( ( M
= ( nat2 @ Z3 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_496_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_497_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_498_nat__less__eq__zless,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_499_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,X: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ X ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_500_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_501_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_502_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_503_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N3: nat] :
( ( I
= ( semiring_1_of_nat @ int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_504_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_505_nat__power__eq,axiom,
! [Z3: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( nat2 @ ( power_power @ int @ Z3 @ N ) )
= ( power_power @ nat @ ( nat2 @ Z3 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_506_log2__of__power__eq,axiom,
! [M: nat,N: nat] :
( ( M
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% log2_of_power_eq
thf(fact_507_even__of__int__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ K ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).
% even_of_int_iff
thf(fact_508_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ M )
= ( ord_less @ int @ W2 @ ( semiring_1_of_nat @ int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_509_nat__dvd__iff,axiom,
! [Z3: int,M: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ Z3 ) @ M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( dvd_dvd @ int @ Z3 @ ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_dvd_iff
thf(fact_510_less__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% less_log2_of_power
thf(fact_511_le__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% le_log2_of_power
thf(fact_512_log2__of__power__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_less
thf(fact_513_even__nat__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat2 @ K ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).
% even_nat_iff
thf(fact_514_arsinh__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% arsinh_0
thf(fact_515_artanh__0,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% artanh_0
thf(fact_516_log__of__power__le,axiom,
! [M: nat,B2: real,N: nat] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_517_log__pow__cancel,axiom,
! [A2: real,B2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ ( power_power @ real @ A2 @ B2 ) )
= ( semiring_1_of_nat @ real @ B2 ) ) ) ) ).
% log_pow_cancel
thf(fact_518_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_519_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_520_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_521_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N2: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% reals_Archimedean2
thf(fact_522_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% real_arch_simple
thf(fact_523_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B5: A] :
( ( A5
= ( zero_zero @ A ) )
=> ( B5
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_524_take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= K )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_525_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_526_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_527_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_528_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_529_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z3: int] :
( ( ( ring_1_of_int @ A @ Z3 )
= ( one_one @ A ) )
= ( Z3
= ( one_one @ int ) ) ) ) ).
% of_int_eq_1_iff
thf(fact_530_of__int__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A @ ( one_one @ int ) )
= ( one_one @ A ) ) ) ).
% of_int_1
thf(fact_531_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_532_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_533_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_534_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ( power_power @ A @ A2 @ M )
= ( power_power @ A @ A2 @ N ) )
= ( M = N ) ) ) ) ).
% power_inject_exp
thf(fact_535_take__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% take_bit_0
thf(fact_536_take__bit__numeral__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_numeral_1
thf(fact_537_log__one,axiom,
! [A2: real] :
( ( log @ A2 @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% log_one
thf(fact_538_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less @ nat @ X @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_539_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_540_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_541_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_542_of__int__1__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% of_int_1_less_iff
thf(fact_543_of__int__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) )
= ( ord_less @ int @ Z3 @ ( one_one @ int ) ) ) ) ).
% of_int_less_1_iff
thf(fact_544_log__eq__one,axiom,
! [A2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ A2 )
= ( one_one @ real ) ) ) ) ).
% log_eq_one
thf(fact_545_log__less__cancel__iff,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_546_log__less__one__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ A2 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_547_one__less__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A2 @ X ) )
= ( ord_less @ real @ A2 @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_548_log__less__zero__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_549_zero__less__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_550_of__nat__nat__take__bit__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,K: int] :
( ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
= ( ring_1_of_int @ A @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_551_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_552_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_553_log__le__cancel__iff,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_554_log__le__one__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ A2 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_555_one__le__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A2 @ X ) )
= ( ord_less_eq @ real @ A2 @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_556_log__le__zero__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_557_zero__le__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_558_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_559_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_560_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: 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 @ M ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ nat @ N @ M ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_561_even__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_take_bit_eq
thf(fact_562_take__bit__of__int,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( ring_1_of_int @ A @ K ) )
= ( ring_1_of_int @ A @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_of_int
thf(fact_563_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_564_take__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) ) ) ) ).
% take_bit_of_nat
thf(fact_565_take__bit__tightened,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ M @ B2 ) ) ) ) ) ).
% take_bit_tightened
thf(fact_566_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_567_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_568_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_569_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A2 ) ) ).
% one_dvd
thf(fact_570_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% unit_imp_dvd
thf(fact_571_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_572_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N: nat,K: int] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_573_take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% take_bit_int_less_eq_self_iff
thf(fact_574_take__bit__nonnegative,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% take_bit_nonnegative
thf(fact_575_take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% take_bit_int_greater_self_iff
thf(fact_576_not__take__bit__negative,axiom,
! [N: nat,K: int] :
~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) ) ).
% not_take_bit_negative
thf(fact_577_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_578_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_579_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_580_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_581_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_582_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_583_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_584_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_585_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_586_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_587_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_588_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_589_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% one_le_power
thf(fact_590_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_591_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_592_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ? [N2: nat] : ( ord_less @ real @ Y @ ( power_power @ real @ X @ N2 ) ) ) ).
% real_arch_pow
thf(fact_593_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_594_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X6: A] :
? [X_12: A] : ( ord_less @ A @ X6 @ X_12 ) ) ).
% linordered_field_no_ub
thf(fact_595_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X6: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X6 ) ) ).
% linordered_field_no_lb
thf(fact_596_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_597_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N4 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_598_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_599_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N4 ) ) ) ) ) ).
% power_increasing
thf(fact_600_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_601_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ? [N2: nat] : ( ord_less @ real @ ( power_power @ real @ X @ N2 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_602_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N4 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_603_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N4 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_604_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_605_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_606_one__power2,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( power_power @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% one_power2
thf(fact_607_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_608_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_609_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ N ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_610_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_611_log__of__power__eq,axiom,
! [M: nat,B2: real,N: nat] :
( ( ( semiring_1_of_nat @ real @ M )
= ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_612_less__log__of__power,axiom,
! [B2: real,N: nat,M: real] :
( ( ord_less @ real @ ( power_power @ real @ B2 @ N ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% less_log_of_power
thf(fact_613_take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_int_less_exp
thf(fact_614_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A2 ) ) ) ).
% take_bit_eq_0_iff
thf(fact_615_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_616_le__log__of__power,axiom,
! [B2: real,N: nat,M: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ B2 @ N ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% le_log_of_power
thf(fact_617_take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% take_bit_int_less_self_iff
thf(fact_618_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_619_log__of__power__less,axiom,
! [M: nat,B2: real,N: nat] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_620_take__bit__int__eq__self,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= K ) ) ) ).
% take_bit_int_eq_self
thf(fact_621_VEBT__internal_Otwo__realpow__ge__two,axiom,
! [N: nat] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N ) )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% VEBT_internal.two_realpow_ge_two
thf(fact_622_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_623_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_624_take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_625_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_626_take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_627_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_628_neg__one__even__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) ) ) ).
% neg_one_even_power
thf(fact_629_neg__one__odd__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% neg_one_odd_power
thf(fact_630_XOR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_631_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_632_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_633_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
= B2 ) ) ).
% verit_minus_simplify(4)
thf(fact_634_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_635_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_636_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_637_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_638_Compl__subset__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_639_Compl__anti__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_640_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_641_Diff__insert0,axiom,
! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B3 ) )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_642_insert__Diff1,axiom,
! [A: $tType,X: A,B3: set @ A,A3: set @ A] :
( ( member @ A @ X @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_643_bit_Oxor__left__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_se5824344971392196577ns_xor @ A @ X @ Y ) )
= Y ) ) ).
% bit.xor_left_self
thf(fact_644_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= N ) ).
% idiff_0_right
thf(fact_645_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_646_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_647_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_648_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_649_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_650_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_651_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_652_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_653_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_654_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add_0
thf(fact_655_double__eq__0__iff,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_eq_0_iff
thf(fact_656_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_657_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_658_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_659_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_660_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V2: num,W2: num,Z3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Z3 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W2 ) ) @ Z3 ) ) ) ).
% add_numeral_left
thf(fact_661_numeral__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% numeral_plus_numeral
thf(fact_662_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_663_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_664_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_665_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_666_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_667_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_668_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A2 ) )
= ( ( zero_zero @ A )
= A2 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_669_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_670_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( A2
= ( uminus_uminus @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_671_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_672_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_673_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_674_neg__numeral__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( M = N ) ) ) ).
% neg_numeral_eq_iff
thf(fact_675_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel
thf(fact_676_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% diff_add_cancel
thf(fact_677_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_678_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_679_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_680_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel_right'
thf(fact_681_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_682_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_683_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_684_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% minus_diff_eq
thf(fact_685_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_triv_right_iff
thf(fact_686_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_triv_left_iff
thf(fact_687_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_add
thf(fact_688_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) )
= ( ord_less @ nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_689_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_690_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_691_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_692_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_693_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% power_one_right
thf(fact_694_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_695_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( ln_ln @ real @ X )
= ( ln_ln @ real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_696_of__int__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: int] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z3 ) )
= ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_minus
thf(fact_697_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ int @ M ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% negative_eq_positive
thf(fact_698_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd @ nat @ M @ ( one_one @ nat ) )
= ( M
= ( one_one @ nat ) ) ) ).
% nat_dvd_1_iff_1
thf(fact_699_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_700_xor__self__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% xor_self_eq
thf(fact_701_xor_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% xor.left_neutral
thf(fact_702_xor_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% xor.right_neutral
thf(fact_703_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zle
thf(fact_704_take__bit__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) ) ) ).
% take_bit_xor
thf(fact_705_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_706_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_707_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_708_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_709_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_710_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_711_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_712_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_713_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_714_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_715_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_716_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_717_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_718_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_719_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_720_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_721_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_le_0_iff_le
thf(fact_722_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_723_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_724_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_725_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_pos
thf(fact_726_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_727_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_0_iff_less
thf(fact_728_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_729_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_730_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ) ).
% le_add_diff_inverse2
thf(fact_731_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% le_add_diff_inverse
thf(fact_732_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_733_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_734_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(168)
thf(fact_735_add__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_736_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_737_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ).
% diff_0
thf(fact_738_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( plus_plus @ A @ A2 @ B2 ) ) ) ).
% diff_minus_eq_add
thf(fact_739_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% uminus_add_conv_diff
thf(fact_740_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_741_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_742_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_less_zero_iff
thf(fact_743_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_744_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ln_ln @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% ln_eq_zero_iff
thf(fact_745_of__int__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z3: int] :
( ( ring_1_of_int @ A @ ( plus_plus @ int @ W2 @ Z3 ) )
= ( plus_plus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_add
thf(fact_746_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_747_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z3: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W2 @ Z3 ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_diff
thf(fact_748_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_749_xor__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_750_xor__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
!= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% xor_negative_int_iff
thf(fact_751_dbl__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% dbl_simps(5)
thf(fact_752_dbl__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_simps(1)
thf(fact_753_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_754_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_755_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_756_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_757_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_758_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_759_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_760_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_le_zero_iff
thf(fact_761_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_762_diff__nat__numeral,axiom,
! [V2: num,V3: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( numeral_numeral @ nat @ V3 ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ V3 ) ) ) ) ).
% diff_nat_numeral
thf(fact_763_zle__add1__eq__le,axiom,
! [W2: int,Z3: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W2 @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_764_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less_eq @ num @ N @ M ) ) ) ).
% neg_numeral_le_iff
thf(fact_765_neg__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less @ num @ N @ M ) ) ) ).
% neg_numeral_less_iff
thf(fact_766_zle__diff1__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq @ int @ W2 @ ( minus_minus @ int @ Z3 @ ( one_one @ int ) ) )
= ( ord_less @ int @ W2 @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_767_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_768_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) )
= ( M != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_769_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( M != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_770_odd__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) )
= ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
!= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ) ).
% odd_add
thf(fact_771_even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_add
thf(fact_772_power2__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A] :
( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_minus
thf(fact_773_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_774_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_775_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_776_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_777_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_plus_one_iff
thf(fact_778_even__diff,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% even_diff
thf(fact_779_power__minus__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_minus_odd
thf(fact_780_Parity_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( power_power @ A @ A2 @ N ) ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_781_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N: nat,Y: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N )
= Y ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_782_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( Y
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_783_nat__numeral__diff__1,axiom,
! [V2: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( one_one @ int ) ) ) ) ).
% nat_numeral_diff_1
thf(fact_784_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_785_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_786_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_787_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_788_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_789_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_790_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_791_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A,J: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_792_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_793_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_794_diff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% diff_add
thf(fact_795_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% le_add_diff
thf(fact_796_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_797_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_798_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_799_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_800_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_801_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_802_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ A2 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_803_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A2 )
= C2 )
= ( B2
= ( plus_plus @ A @ C2 @ A2 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_804_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_805_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_806_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A2 )
= B2 ) ) ) ).
% add.inverse_unique
thf(fact_807_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_808_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add_eq_0_iff
thf(fact_809_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A] :
( ~ ( ord_less @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_810_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( ord_less @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_811_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% less_diff_eq
thf(fact_812_of__nat__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se5824344971392196577ns_xor @ nat @ M @ N ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_xor_eq
thf(fact_813_add__diff__add,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A,D4: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D4 ) )
= ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( minus_minus @ A @ C2 @ D4 ) ) ) ) ).
% add_diff_add
thf(fact_814_minus__diff__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% minus_diff_minus
thf(fact_815_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
=> ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_816_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_817_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_818_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_819_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: A,K: A,B2: A,A2: A] :
( ( B3
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_820_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K: A,A2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( uminus_uminus @ A @ A3 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_821_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( minus_minus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_822_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B3: A,K: A,B2: A,A2: A] :
( ( B3
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A2 @ B3 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_823_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= C2 )
= ( A2
= ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_824_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( A2
= ( minus_minus @ A @ C2 @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_825_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_826_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_827_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_828_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_829_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_830_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D4 ) )
=> ( ( A2 = B2 )
= ( C2 = D4 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_831_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_832_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B5: A] : ( plus_plus @ A @ B5 @ A5 ) ) ) ) ).
% add.commute
thf(fact_833_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_834_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_835_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B5: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_836_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% minus_diff_commute
thf(fact_837_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_838_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_839_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B5: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_840_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_841_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_842_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_843_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ( plus_plus @ A @ C2 @ B2 )
= A2 )
=> ( C2
= ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_844_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_845_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% diff_diff_eq
thf(fact_846_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_847_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_848_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ X2 @ X2 ) ) ) ) ).
% dbl_def
thf(fact_849_xor_Oassoc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) @ C2 )
= ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( bit_se5824344971392196577ns_xor @ A @ B2 @ C2 ) ) ) ) ).
% xor.assoc
thf(fact_850_xor_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [A5: A,B5: A] : ( bit_se5824344971392196577ns_xor @ A @ B5 @ A5 ) ) ) ) ).
% xor.commute
thf(fact_851_xor_Oleft__commute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ B2 @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ C2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( bit_se5824344971392196577ns_xor @ A @ B2 @ C2 ) ) ) ) ).
% xor.left_commute
thf(fact_852_of__int__xor__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,L: int] :
( ( ring_1_of_int @ A @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) ) ) ).
% of_int_xor_eq
thf(fact_853_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus @ int @ ( zero_zero @ int ) @ L )
= ( uminus_uminus @ int @ L ) ) ).
% minus_int_code(2)
thf(fact_854_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) @ Z3 )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_855_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_856_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less @ int @ I @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I2: int] :
( ( ord_less @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_857_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_858_zless__add1__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W2 @ Z3 )
| ( W2 = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_859_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I2: int] :
( ( ord_less @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_860_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_861_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_862_of__nat__diff,axiom,
! [A: $tType] :
( ( semiring_1_cancel @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N ) )
= ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% of_nat_diff
thf(fact_863_int__minus,axiom,
! [N: nat,M: nat] :
( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ N @ M ) )
= ( semiring_1_of_nat @ int @ ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M ) ) ) ) ) ).
% int_minus
thf(fact_864_odd__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) @ Z3 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ Z3 @ ( zero_zero @ int ) ) ) ).
% odd_less_0_iff
thf(fact_865_take__bit__nat__less__eq__self,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M ) ).
% take_bit_nat_less_eq_self
thf(fact_866_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q3 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N @ Q3 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_867_zless__imp__add1__zle,axiom,
! [W2: int,Z3: int] :
( ( ord_less @ int @ W2 @ Z3 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_868_add1__zle__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z3 )
= ( ord_less @ int @ W2 @ Z3 ) ) ).
% add1_zle_eq
thf(fact_869_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( zero_zero @ int ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
= ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( one_one @ int ) ) ) ) ).
% take_bit_decr_eq
thf(fact_870_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( minus_minus @ A @ A5 @ B5 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_871_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D4: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D4 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D4 ) ) ) ) ) ).
% diff_mono
thf(fact_872_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_873_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_874_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D4 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C2 @ D4 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_875_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D4: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D4 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D4 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_876_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D4 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C2 @ D4 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_877_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_878_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_879_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_minus_iff
thf(fact_880_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_le_iff
thf(fact_881_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_imp_neg_le
thf(fact_882_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_883_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% less_minus_iff
thf(fact_884_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_less_iff
thf(fact_885_neg__numeral__neq__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
!= ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_neq_numeral
thf(fact_886_numeral__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( numeral_numeral @ A @ M )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_887_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_888_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_889_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_890_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_891_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_892_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_893_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_894_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D4 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D4 ) ) ) ) ) ).
% add_mono
thf(fact_895_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_896_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C5: A] :
( B2
!= ( plus_plus @ A @ A2 @ C5 ) ) ) ) ).
% less_eqE
thf(fact_897_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_898_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
? [C6: A] :
( B5
= ( plus_plus @ A @ A5 @ C6 ) ) ) ) ) ).
% le_iff_add
thf(fact_899_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_900_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_901_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_902_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_903_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_904_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_905_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D4 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D4 ) ) ) ) ) ).
% add_strict_mono
thf(fact_906_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_907_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_908_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_909_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_910_dvd__diff__commute,axiom,
! [A: $tType] :
( ( euclid5891614535332579305n_ring @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% dvd_diff_commute
thf(fact_911_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_912_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_913_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N @ M )
= ( zero_zero @ nat ) )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_914_double__diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ( minus_minus @ ( set @ A ) @ B3 @ ( minus_minus @ ( set @ A ) @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_915_Diff__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ).
% Diff_subset
thf(fact_916_Diff__mono,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,D: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ D @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_917_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ord_less @ nat @ M @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_918_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_919_dvd__add__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_add_right_iff
thf(fact_920_dvd__add__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_add_left_iff
thf(fact_921_dvd__add,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ C2 )
=> ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ).
% dvd_add
thf(fact_922_insert__Diff__if,axiom,
! [A: $tType,X: A,B3: set @ A,A3: set @ A] :
( ( ( member @ A @ X @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) )
& ( ~ ( member @ A @ X @ B3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B3 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_923_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_924_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_925_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_926_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_927_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_928_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_929_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq @ nat @ A2 @ C2 )
=> ( ( ord_less_eq @ nat @ B2 @ C2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_930_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_931_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_932_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% plus_int_code(1)
thf(fact_933_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
= L ) ).
% plus_int_code(2)
thf(fact_934_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X ) ) @ ( uminus_uminus @ real @ X ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_935_int__diff__cases,axiom,
! [Z3: int] :
~ ! [M4: nat,N2: nat] :
( Z3
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_936_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ M )
=> ( ( dvd_dvd @ nat @ K @ N )
=> ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_937_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A5: nat,B5: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_938_int__cases2,axiom,
! [Z3: int] :
( ! [N2: nat] :
( Z3
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( Z3
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_939_take__bit__add,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% take_bit_add
thf(fact_940_psubset__imp__ex__mem,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ? [B6: A] : ( member @ A @ B6 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_941_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd @ int @ K @ ( minus_minus @ int @ M @ N ) )
=> ( ( dvd_dvd @ int @ K @ N )
=> ( dvd_dvd @ int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_942_uminus__dvd__conv_I1_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D5: int] : ( dvd_dvd @ int @ ( uminus_uminus @ int @ D5 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_943_uminus__dvd__conv_I2_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D5: int,T3: int] : ( dvd_dvd @ int @ D5 @ ( uminus_uminus @ int @ T3 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_944_take__bit__diff,axiom,
! [N: nat,K: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ L ) ) ) ).
% take_bit_diff
thf(fact_945_take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% take_bit_minus
thf(fact_946_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ln_ln @ real @ X )
= ( minus_minus @ real @ X @ ( one_one @ real ) ) )
=> ( X
= ( one_one @ real ) ) ) ) ).
% ln_eq_minus_one
thf(fact_947_zero__one__enat__neq_I1_J,axiom,
( ( zero_zero @ extended_enat )
!= ( one_one @ extended_enat ) ) ).
% zero_one_enat_neq(1)
thf(fact_948_of__int__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% of_int_neg_numeral
thf(fact_949_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 ) ) ) )
& ! [Y2: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y2 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y2 @ ( one_one @ int ) ) ) ) )
=> ( Y2 = X3 ) ) ) ) ).
% floor_exists1
thf(fact_950_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_951_int__ops_I6_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_952_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( minus_minus @ int @ X @ Y ) )
= ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_953_nat__diff__distrib,axiom,
! [Z6: int,Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
=> ( ( ord_less_eq @ int @ Z6 @ Z3 )
=> ( ( nat2 @ ( minus_minus @ int @ Z3 @ Z6 ) )
= ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_954_le__imp__0__less,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% le_imp_0_less
thf(fact_955_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_956_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( minus_minus @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_le_minus_one
thf(fact_957_ln__less__self,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_958_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_959_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y ) ) )
= ( ( ( ord_less_eq @ nat @ Y @ X )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y ) ) ) )
& ( ( ord_less @ nat @ X @ Y )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_960_XOR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) ) ) ) ).
% XOR_lower
thf(fact_961_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_962_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_963_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_964_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_le_numeral
thf(fact_965_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_966_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_less_numeral
thf(fact_967_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_968_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_969_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_970_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_971_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_972_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_973_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_974_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_975_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_976_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_977_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_978_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_979_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_980_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_981_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_982_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_983_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ! [C5: A] :
( ( B2
= ( plus_plus @ A @ A2 @ C5 ) )
=> ( C5
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_984_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_985_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_986_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_987_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_988_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D4 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D4 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_989_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D4 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D4 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_990_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_991_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_992_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_993_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_994_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_995_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_996_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D4: B,S: B] :
? [Z: B] :
! [X6: B] :
( ( ord_less @ B @ X6 @ Z )
=> ( ( ~ ( dvd_dvd @ B @ D4 @ ( plus_plus @ B @ X6 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D4 @ ( plus_plus @ B @ X6 @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_997_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D4: B,S: B] :
? [Z: B] :
! [X6: B] :
( ( ord_less @ B @ X6 @ Z )
=> ( ( dvd_dvd @ B @ D4 @ ( plus_plus @ B @ X6 @ S ) )
= ( dvd_dvd @ B @ D4 @ ( plus_plus @ B @ X6 @ S ) ) ) ) ) ).
% minf(9)
thf(fact_998_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D4: B,S: B] :
? [Z: B] :
! [X6: B] :
( ( ord_less @ B @ Z @ X6 )
=> ( ( ~ ( dvd_dvd @ B @ D4 @ ( plus_plus @ B @ X6 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D4 @ ( plus_plus @ B @ X6 @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_999_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D4: B,S: B] :
? [Z: B] :
! [X6: B] :
( ( ord_less @ B @ Z @ X6 )
=> ( ( dvd_dvd @ B @ D4 @ ( plus_plus @ B @ X6 @ S ) )
= ( dvd_dvd @ B @ D4 @ ( plus_plus @ B @ X6 @ S ) ) ) ) ) ).
% pinf(9)
thf(fact_1000_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1001_subset__Diff__insert,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X: A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ ( insert @ A @ X @ C3 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ C3 ) )
& ~ ( member @ A @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1002_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1003_diff__less__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_less_eq @ nat @ C2 @ A2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1004_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_1005_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ M ) )
= ( ( ord_less @ nat @ N @ M )
| ( dvd_dvd @ nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_1006_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
=> ( ( dvd_dvd @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_1007_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
=> ( ( dvd_dvd @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1008_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ M @ N )
= ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1009_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1010_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W3: int,Z7: int] :
? [N3: nat] :
( Z7
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1011_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_1012_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_1013_of__int__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K3: int] : ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K3 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K3 ) ) ) ) ) ) ).
% of_int_of_nat
thf(fact_1014_ln__bound,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_bound
thf(fact_1015_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1016_ln__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% ln_less_zero
thf(fact_1017_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) ) ) ).
% ln_gt_zero
thf(fact_1018_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) ) ) ).
% ln_ge_zero
thf(fact_1019_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_1020_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_1021_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_1022_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_1023_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_1024_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_1025_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_1026_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_1027_neg__numeral__le__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_le_one
thf(fact_1028_neg__one__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% neg_one_le_numeral
thf(fact_1029_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% neg_numeral_le_neg_one
thf(fact_1030_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_le_neg_one
thf(fact_1031_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1032_neg__numeral__less__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_less_one
thf(fact_1033_neg__one__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% neg_one_less_numeral
thf(fact_1034_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_less_neg_one
thf(fact_1035_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1036_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1037_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_1038_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_1039_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_1040_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_1041_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_1042_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_1043_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y @ E ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_epsilon
thf(fact_1044_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_1045_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_1046_power__minus__Bit0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) )
= ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ) ) ).
% power_minus_Bit0
thf(fact_1047_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( M
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_1048_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% int_zle_neg
thf(fact_1049_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( zero_zero @ int ) ) ).
% negative_zle_0
thf(fact_1050_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1051_take__bit__numeral__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_1052_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ Z3 )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1053_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1054_nat__take__bit__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) ) ) ) ).
% nat_take_bit_eq
thf(fact_1055_take__bit__nat__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_nat_eq
thf(fact_1056_power2__commute,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ ( minus_minus @ A @ Y @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_commute
thf(fact_1057_even__minus,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ A @ A2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ).
% even_minus
thf(fact_1058_even__xor__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_xor_iff
thf(fact_1059_power2__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [X: A,Y: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% power2_eq_iff
thf(fact_1060_odd__even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% odd_even_add
thf(fact_1061_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_1062_int__cases3,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ! [N2: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% int_cases3
thf(fact_1063_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_1064_take__bit__nat__eq__self,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M )
= M ) ) ).
% take_bit_nat_eq_self
thf(fact_1065_take__bit__nat__less__exp,axiom,
! [N: nat,M: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_nat_less_exp
thf(fact_1066_take__bit__nat__eq__self__iff,axiom,
! [N: nat,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M )
= M )
= ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_1067_power2__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A2: A] :
( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( A2
= ( one_one @ A ) )
| ( A2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% power2_eq_1_iff
thf(fact_1068_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,M: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) )
!= ( zero_zero @ A ) ) ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1069_uminus__power__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A2: A] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( power_power @ A @ A2 @ N ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% uminus_power_if
thf(fact_1070_realpow__square__minus__le,axiom,
! [U2: real,X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( power_power @ real @ U2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% realpow_square_minus_le
thf(fact_1071_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1072_neg__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% neg_int_cases
thf(fact_1073_take__bit__nat__less__self__iff,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M ) ) ).
% take_bit_nat_less_self_iff
thf(fact_1074_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_1075_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_1076_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_1077_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_1078_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_1079_minus__one__power__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% minus_one_power_iff
thf(fact_1080_ln__2__less__1,axiom,
ord_less @ real @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ).
% ln_2_less_1
thf(fact_1081_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% compl_less_compl_iff
thf(fact_1082_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_1083_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_1084_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N3: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I4: A] : ( plus_plus @ A @ I4 @ ( one_one @ A ) )
@ N3
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_1085_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1086_Bolzano,axiom,
! [A2: real,B2: real,P: real > real > $o] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [A6: real,B6: real,C5: real] :
( ( P @ A6 @ B6 )
=> ( ( P @ B6 @ C5 )
=> ( ( ord_less_eq @ real @ A6 @ B6 )
=> ( ( ord_less_eq @ real @ B6 @ C5 )
=> ( P @ A6 @ C5 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [A6: real,B6: real] :
( ( ( ord_less_eq @ real @ A6 @ X3 )
& ( ord_less_eq @ real @ X3 @ B6 )
& ( ord_less @ real @ ( minus_minus @ real @ B6 @ A6 ) @ D2 ) )
=> ( P @ A6 @ B6 ) ) ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Bolzano
thf(fact_1087_DiffI,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_1088_Diff__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C2 @ A3 )
& ~ ( member @ A @ C2 @ B3 ) ) ) ).
% Diff_iff
thf(fact_1089_Diff__idemp,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ).
% Diff_idemp
thf(fact_1090_ComplI,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% ComplI
thf(fact_1091_Compl__iff,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( ~ ( member @ A @ C2 @ A3 ) ) ) ).
% Compl_iff
thf(fact_1092_Compl__eq__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A3 )
= ( uminus_uminus @ ( set @ A ) @ B3 ) )
= ( A3 = B3 ) ) ).
% Compl_eq_Compl_iff
thf(fact_1093_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_1094_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_right
thf(fact_1095_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( divide_divide @ A @ C2 @ A2 )
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_left
thf(fact_1096_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_1097_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_1098_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_1099_bits__div__by__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_1100_bits__div__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_1101_bits__div__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% bits_div_by_1
thf(fact_1102_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% div_by_1
thf(fact_1103_div__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ).
% div_minus_minus
thf(fact_1104_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_1105_div__dvd__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( divide_divide @ A @ C2 @ A2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ) ).
% div_dvd_div
thf(fact_1106_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_1107_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_1108_zdiv__numeral__Bit0,axiom,
! [V2: num,W2: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit0 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W2 ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_1109_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1110_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1111_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1112_real__add__minus__iff,axiom,
! [X: real,A2: real] :
( ( ( plus_plus @ real @ X @ ( uminus_uminus @ real @ A2 ) )
= ( zero_zero @ real ) )
= ( X = A2 ) ) ).
% real_add_minus_iff
thf(fact_1113_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_1114_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_1115_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_1116_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B2 @ A2 ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_1117_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ A2 )
= ( one_one @ A ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_1118_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( A2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) )
& ( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_1119_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_1120_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A2 @ B2 ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_1121_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_1122_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_1123_div__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ A2 ) ) ) ).
% div_minus1_right
thf(fact_1124_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= A2 ) ) ) ).
% unit_div_1_div_1
thf(fact_1125_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_1126_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_1127_div__add,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_add
thf(fact_1128_div__diff,axiom,
! [A: $tType] :
( ( idom_modulo @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_diff
thf(fact_1129_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_1130_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1131_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1132_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1133_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_pos_pos_trivial
thf(fact_1134_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_neg_neg_trivial
thf(fact_1135_semiring__norm_I2_J,axiom,
( ( plus_plus @ num @ one2 @ one2 )
= ( bit0 @ one2 ) ) ).
% semiring_norm(2)
thf(fact_1136_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_1137_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_1138_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_1139_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1140_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1141_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1142_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1143_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_1144_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= M ) ).
% add_self_div_2
thf(fact_1145_diff__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_1146_diff__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% diff_numeral_simps(2)
thf(fact_1147_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1148_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1149_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1150_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1151_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_1152_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% half_negative_int_iff
thf(fact_1153_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_1154_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_1155_minus__1__div__2__eq,axiom,
! [A: $tType] :
( ( euclid8789492081693882211th_nat @ A )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% minus_1_div_2_eq
thf(fact_1156_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ one2 ) ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_1157_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_1158_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_1159_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_1160_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_1161_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) )
= ( ( ord_less @ nat @ M @ N )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_1162_minus__real__def,axiom,
( ( minus_minus @ real )
= ( ^ [X2: real,Y4: real] : ( plus_plus @ real @ X2 @ ( uminus_uminus @ real @ Y4 ) ) ) ) ).
% minus_real_def
thf(fact_1163_DiffE,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% DiffE
thf(fact_1164_ComplD,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
=> ~ ( member @ A @ C2 @ A3 ) ) ).
% ComplD
thf(fact_1165_DiffD1,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
=> ( member @ A @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_1166_DiffD2,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ( member @ A @ C2 @ B3 ) ) ).
% DiffD2
thf(fact_1167_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ ^ [X2: A] : ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1168_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A4: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1169_double__complement,axiom,
! [A: $tType,A3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_1170_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1171_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1172_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1173_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1174_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1175_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
= ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_1176_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) ) ).
% real_of_int_div4
thf(fact_1177_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_1178_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A4: set @ A] :
( collect @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_1179_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
& ~ ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1180_zdiv__int,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A2 @ B2 ) )
= ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zdiv_int
thf(fact_1181_real__of__int__div,axiom,
! [D4: int,N: int] :
( ( dvd_dvd @ int @ D4 @ N )
=> ( ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ D4 ) )
= ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ D4 ) ) ) ) ).
% real_of_int_div
thf(fact_1182_real__of__int__div2,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) ) ) ).
% real_of_int_div2
thf(fact_1183_real__of__int__div3,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_int_div3
thf(fact_1184_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1185_real__of__nat__div,axiom,
! [D4: nat,N: nat] :
( ( dvd_dvd @ nat @ D4 @ N )
=> ( ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ D4 ) )
= ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ D4 ) ) ) ) ).
% real_of_nat_div
thf(fact_1186_div__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% div_minus_right
thf(fact_1187_power__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ A2 @ B2 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_divide
thf(fact_1188_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_1189_dvd__div__eq__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( ( divide_divide @ A @ A2 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1190_dvd__div__eq__cancel,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
=> ( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( A2 = B2 ) ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1191_div__div__div__same,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [D4: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ D4 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ D4 ) @ ( divide_divide @ A @ B2 @ D4 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_div_div_same
thf(fact_1192_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1193_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1194_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
= ( P @ B6 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ ( zero_zero @ nat ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
=> ( P @ A6 @ ( plus_plus @ nat @ A6 @ B6 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1195_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= M )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_1196_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1197_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_1198_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1199_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1200_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1201_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1202_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1203_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1204_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1205_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
=> ( ord_less @ nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1206_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less @ nat @ M @ N )
=> ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1207_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1208_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus @ nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1209_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1210_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1211_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1212_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1213_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus @ nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1214_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ).
% add_leD2
thf(fact_1215_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% add_leD1
thf(fact_1216_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_1217_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_1218_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M @ N )
=> ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1219_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( minus_minus @ nat @ J @ I )
= K )
= ( J
= ( plus_plus @ nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1220_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K )
= ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1221_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
= ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1222_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1223_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1224_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A5: nat,B5: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_1225_iadd__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_plus @ extended_enat @ M @ N )
= ( zero_zero @ extended_enat ) )
= ( ( M
= ( zero_zero @ extended_enat ) )
& ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% iadd_is_0
thf(fact_1226_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1227_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1228_div__neg__pos__less0,axiom,
! [A2: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_neg_pos_less0
thf(fact_1229_add__diff__assoc__enat,axiom,
! [Z3: extended_enat,Y: extended_enat,X: extended_enat] :
( ( ord_less_eq @ extended_enat @ Z3 @ Y )
=> ( ( plus_plus @ extended_enat @ X @ ( minus_minus @ extended_enat @ Y @ Z3 ) )
= ( minus_minus @ extended_enat @ ( plus_plus @ extended_enat @ X @ Y ) @ Z3 ) ) ) ).
% add_diff_assoc_enat
thf(fact_1230_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_1231_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_nat_div3
thf(fact_1232_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_1233_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_1234_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ).
% div_exp_eq
thf(fact_1235_div__positive,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_positive
thf(fact_1236_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_1237_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1238_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_1239_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_1240_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_1241_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_1242_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1243_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_right_mono
thf(fact_1244_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_1245_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_1246_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1247_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1248_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) )
& ( C2
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_1249_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_1250_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_1251_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_1252_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_1253_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_1254_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( A2 = B2 ) ) ) ) ).
% right_inverse_eq
thf(fact_1255_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
= A2 ) ) ).
% divide_numeral_1
thf(fact_1256_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_1257_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_1258_dvd__div__eq__0__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1259_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_one_over
thf(fact_1260_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_1261_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_1262_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_div_cancel
thf(fact_1263_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_1264_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1265_dvd__neg__div,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% dvd_neg_div
thf(fact_1266_dvd__div__neg,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% dvd_div_neg
thf(fact_1267_div__power,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,N: nat] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( power_power @ A @ ( divide_divide @ A @ A2 @ B2 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% div_power
thf(fact_1268_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( ord_less @ nat @ M @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1269_ln__div,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( divide_divide @ real @ X @ Y ) )
= ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_1270_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ( ord_less_eq @ int @ B2 @ A2 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1271_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1272_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1273_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K ) )
= ( ord_less_eq @ int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1274_div__nonpos__pos__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1275_div__nonneg__neg__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1276_div__positive__int,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ L @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) ) ) ).
% div_positive_int
thf(fact_1277_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) )
= ( ( K
= ( zero_zero @ int ) )
| ( L
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
| ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% div_int_pos_iff
thf(fact_1278_zdiv__mono2__neg,axiom,
! [A2: int,B8: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B8 )
=> ( ( ord_less_eq @ int @ B8 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B8 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1279_zdiv__mono1__neg,axiom,
! [A2: int,A7: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A7 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A7 @ B2 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1280_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide @ int @ I @ K )
= ( zero_zero @ int ) )
= ( ( K
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
& ( ord_less @ int @ I @ K ) )
| ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
& ( ord_less @ int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1281_zdiv__mono2,axiom,
! [A2: int,B8: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B8 )
=> ( ( ord_less_eq @ int @ B8 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B8 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1282_zdiv__mono1,axiom,
! [A2: int,A7: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A7 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_1283_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ Y ) )
= ( ord_less @ real @ ( uminus_uminus @ real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1284_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( plus_plus @ real @ X @ Y ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ Y @ ( uminus_uminus @ real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1285_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1286_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_1287_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ Y ) )
= ( ord_less_eq @ real @ ( uminus_uminus @ real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1288_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ X @ Y ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ Y @ ( uminus_uminus @ real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_1289_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ( plus_plus @ nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1290_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less @ nat @ A2 @ B2 )
& ~ ( P @ ( zero_zero @ nat ) ) )
| ? [D5: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1291_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ( ( ord_less @ nat @ A2 @ B2 )
=> ( P @ ( zero_zero @ nat ) ) )
& ! [D5: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1292_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less @ nat @ M4 @ N2 )
=> ( ord_less @ nat @ ( F2 @ M4 ) @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1293_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,N: nat] :
( ( P @ K )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ? [Y2: A] :
( ( P @ Y2 )
& ~ ( ord_less_eq @ nat @ ( F2 @ Y2 ) @ ( F2 @ X3 ) ) ) )
=> ? [Y3: A] :
( ( P @ Y3 )
& ~ ( ord_less @ nat @ ( F2 @ Y3 ) @ ( plus_plus @ nat @ ( F2 @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_1294_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1295_int__ops_I5_J,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A2 @ B2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_1296_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N @ M ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% int_plus
thf(fact_1297_zadd__int__left,axiom,
! [M: nat,N: nat,Z3: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N ) ) @ Z3 ) ) ).
% zadd_int_left
thf(fact_1298_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_1299_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_1300_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_1301_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_1302_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_1303_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_1304_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_1305_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_1306_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_1307_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ A2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A2 @ B2 ) )
| ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_1308_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_1309_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_1310_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_1311_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% gt_half_sum
thf(fact_1312_numeral__Bit0__div__2,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% numeral_Bit0_div_2
thf(fact_1313_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_1314_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1315_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N ) )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_diff
thf(fact_1316_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,I: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
= I ) ) ).
% of_nat_aux.simps(1)
thf(fact_1317_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) )
= ( ( ord_less_eq @ nat @ N @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1318_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1319_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ( divide_divide @ nat @ M @ N )
= M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_1320_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1321_ln__diff__le,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X @ Y ) @ Y ) ) ) ) ).
% ln_diff_le
thf(fact_1322_ln__add__one__self__le__self,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self
thf(fact_1323_ln__add__one__self__le__self2,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self2
thf(fact_1324_verit__less__mono__div__int2,axiom,
! [A3: int,B3: int,N: int] :
( ( ord_less_eq @ int @ A3 @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B3 @ N ) @ ( divide_divide @ int @ A3 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1325_int__le__real__less,axiom,
( ( ord_less_eq @ int )
= ( ^ [N3: int,M3: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N3 ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M3 ) @ ( one_one @ real ) ) ) ) ) ).
% int_le_real_less
thf(fact_1326_int__less__real__le,axiom,
( ( ord_less @ int )
= ( ^ [N3: int,M3: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N3 ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M3 ) ) ) ) ).
% int_less_real_le
thf(fact_1327_xor__nat__def,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se5824344971392196577ns_xor @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% xor_nat_def
thf(fact_1328_nat__int__add,axiom,
! [A2: nat,B2: nat] :
( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
= ( plus_plus @ nat @ A2 @ B2 ) ) ).
% nat_int_add
thf(fact_1329_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A5: nat,B5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_1330_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_1331_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ A2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A2 @ B2 ) )
| ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_1332_field__sum__of__halves,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( plus_plus @ A @ ( divide_divide @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= X ) ) ).
% field_sum_of_halves
thf(fact_1333_bit__eq__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A5: A,B5: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) )
& ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ B5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_1334_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
= ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1335_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_1336_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1337_nat__less__real__le,axiom,
( ( ord_less @ nat )
= ( ^ [N3: nat,M3: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M3 ) ) ) ) ).
% nat_less_real_le
thf(fact_1338_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N3: nat,M3: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M3 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_1339_log__base__change,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ B2 @ X )
= ( divide_divide @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ B2 ) ) ) ) ) ).
% log_base_change
thf(fact_1340_log__divide,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A2 @ ( divide_divide @ real @ X @ Y ) )
= ( minus_minus @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ).
% log_divide
thf(fact_1341_nat__add__distrib,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
=> ( ( nat2 @ ( plus_plus @ int @ Z3 @ Z6 ) )
= ( plus_plus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1342_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_1343_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% half_gt_zero_iff
thf(fact_1344_half__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% half_gt_zero
thf(fact_1345_field__less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ X @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% field_less_half_sum
thf(fact_1346_inverse__of__nat__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_1347_exp__add__not__zero__imp__right,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_right
thf(fact_1348_exp__add__not__zero__imp__left,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_left
thf(fact_1349_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1350_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% minus_1_div_exp_eq_int
thf(fact_1351_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ K @ L )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1352_log__base__pow,axiom,
! [A2: real,N: nat,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ ( power_power @ real @ A2 @ N ) @ X )
= ( divide_divide @ real @ ( log @ A2 @ X ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log_base_pow
thf(fact_1353_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_1354_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_1355_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_1356_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_less_swap2
thf(fact_1357_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_less_swap1
thf(fact_1358_ln__one__plus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ X @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) ) ) ) ).
% ln_one_plus_pos_lower_bound
thf(fact_1359_ex__power__ivl2,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1360_ex__power__ivl1,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ? [N2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1361_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% even_mask_div_iff'
thf(fact_1362_artanh__def,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% artanh_def
thf(fact_1363_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% even_mask_div_iff
thf(fact_1364_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_1365_real__average__minus__second,axiom,
! [B2: real,A2: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ B2 @ A2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A2 )
= ( divide_divide @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_second
thf(fact_1366_real__average__minus__first,axiom,
! [A2: real,B2: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ A2 @ B2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A2 )
= ( divide_divide @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_first
thf(fact_1367_VEBT__internal_Opower__minus__is__div,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq @ nat @ B2 @ A2 )
=> ( ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% VEBT_internal.power_minus_is_div
thf(fact_1368_VEBT__internal_Opow__sum,axiom,
! [A2: nat,B2: nat] :
( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ).
% VEBT_internal.pow_sum
thf(fact_1369_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_1370_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_1371_ceiling__log__nat__eq__powr__iff,axiom,
! [B2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) )
= ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_1372_ceiling__log__nat__eq__if,axiom,
! [B2: nat,N: nat,K: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
=> ( ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_1373_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_1374_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_1375_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_1376_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_1377_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_1378_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_1379_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% numeral_times_numeral
thf(fact_1380_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [V2: num,W2: num,Z3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Z3 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W2 ) ) @ Z3 ) ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1381_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_1382_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult_1
thf(fact_1383_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% mult_minus_left
thf(fact_1384_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( times_times @ A @ A2 @ B2 ) ) ) ).
% minus_mult_minus
thf(fact_1385_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% mult_minus_right
thf(fact_1386_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mult
thf(fact_1387_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_1388_real__divide__square__eq,axiom,
! [R3: real,A2: real] :
( ( divide_divide @ real @ ( times_times @ real @ R3 @ A2 ) @ ( times_times @ real @ R3 @ R3 ) )
= ( divide_divide @ real @ A2 @ R3 ) ) ).
% real_divide_square_eq
thf(fact_1389_of__int__ceiling__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) )
= X )
= ( ? [N3: int] :
( X
= ( ring_1_of_int @ A @ N3 ) ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_1390_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_1391_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,A2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_1392_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_1393_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A2: A,C2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_1394_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_1395_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A2: A,B2: A,V2: num] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_1396_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V2: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C2 ) ) ) ) ).
% distrib_left_numeral
thf(fact_1397_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1398_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1399_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_1400_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_1401_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1402_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1403_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1404_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1405_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1406_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1407_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V2: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C2 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1408_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A2: A,B2: A,V2: num] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1409_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1410_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1411_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1412_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(170)
thf(fact_1413_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(171)
thf(fact_1414_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W2 ) ) @ Y ) ) ) ).
% semiring_norm(172)
thf(fact_1415_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1416_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1417_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1418_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1419_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_1420_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_1421_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_1422_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A2 ) @ B2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1423_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1424_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% dvd_div_mult_self
thf(fact_1425_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ A2 ) )
= B2 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_1426_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_1427_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_1428_ceiling__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% ceiling_numeral
thf(fact_1429_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_1430_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_1431_round__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ int @ N ) ) ) ).
% round_numeral
thf(fact_1432_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_1433_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) )
= A2 )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1434_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1435_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_1436_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_1437_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_1438_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_1439_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1440_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1441_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_1442_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_1443_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_1444_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_1445_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_1446_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A2: 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 ) @ A2 ) )
= A2 ) ) ).
% left_minus_one_mult_self
thf(fact_1447_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( divide_divide @ A @ B2 @ A2 ) ) ) ) ).
% unit_mult_div_div
thf(fact_1448_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% unit_div_mult_self
thf(fact_1449_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N: num] :
( ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ N ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% power_add_numeral
thf(fact_1450_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N: num,B2: A] :
( ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% power_add_numeral2
thf(fact_1451_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= A2 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1452_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1453_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_1454_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_1455_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_1456_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_1457_even__mult__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_mult_iff
thf(fact_1458_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_1459_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_1460_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_1461_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% ceiling_le_numeral
thf(fact_1462_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V2 ) @ X ) ) ) ).
% numeral_less_ceiling
thf(fact_1463_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_1464_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_1465_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_1466_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_1467_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( numeral_numeral @ A @ V2 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_add_numeral
thf(fact_1468_ceiling__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_neg_numeral
thf(fact_1469_ceiling__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_diff_numeral
thf(fact_1470_ceiling__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: num,N: nat] :
( ( archimedean_ceiling @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% ceiling_numeral_power
thf(fact_1471_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_1472_nat__ceiling__le__eq,axiom,
! [X: real,A2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A2 )
= ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1473_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_1474_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_1475_ceiling__divide__eq__div__numeral,axiom,
! [A2: num,B2: num] :
( ( archimedean_ceiling @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ).
% ceiling_divide_eq_div_numeral
thf(fact_1476_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_1477_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_le_ceiling
thf(fact_1478_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_1479_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_1480_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_1481_ceiling__minus__divide__eq__div__numeral,axiom,
! [A2: num,B2: num] :
( ( archimedean_ceiling @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ).
% ceiling_minus_divide_eq_div_numeral
thf(fact_1482_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A2 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_1483_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_1484_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_1485_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_1486_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_1487_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1488_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_1489_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A5: A,B5: A] : ( times_times @ A @ B5 @ A5 ) ) ) ) ).
% mult.commute
thf(fact_1490_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.left_commute
thf(fact_1491_mult__commute__abs,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [C2: A] :
( ( ^ [X2: A] : ( times_times @ A @ X2 @ C2 ) )
= ( times_times @ A @ C2 ) ) ) ).
% mult_commute_abs
thf(fact_1492_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A2 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A2 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_1493_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_1494_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_1495_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_1496_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_1497_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_1498_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1499_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.comm_neutral
thf(fact_1500_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1501_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1502_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1503_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_1504_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_1505_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,E2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E2 ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_1506_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% left_diff_distrib
thf(fact_1507_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% right_diff_distrib
thf(fact_1508_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A2 )
= ( minus_minus @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) ) ) ) ).
% left_diff_distrib'
thf(fact_1509_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% right_diff_distrib'
thf(fact_1510_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
=> ! [X6: A,K4: A] :
( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D ) ) )
| ( Q @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1511_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D ) ) ) )
=> ! [X6: A,K4: A] :
( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D ) ) )
& ( Q @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1512_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ A2 )
= ( times_times @ A @ B2 @ B2 ) )
= ( ( A2 = B2 )
| ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% square_eq_iff
thf(fact_1513_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_mult_commute
thf(fact_1514_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ A2 @ N ) @ A2 )
= ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_commutes
thf(fact_1515_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( power_power @ A @ ( times_times @ A @ A2 @ B2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_mult_distrib
thf(fact_1516_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_1517_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) ) ) ).
% dvd_triv_right
thf(fact_1518_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ B2 @ C2 ) ) ) ).
% dvd_mult_right
thf(fact_1519_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D4 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D4 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_1520_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) ) ) ).
% dvd_triv_left
thf(fact_1521_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ A2 @ C2 ) ) ) ).
% dvd_mult_left
thf(fact_1522_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult2
thf(fact_1523_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
=> ? [B9: A,C7: A] :
( ( A2
= ( times_times @ A @ B9 @ C7 ) )
& ( dvd_dvd @ A @ B9 @ B2 )
& ( dvd_dvd @ A @ C7 @ C2 ) ) ) ) ).
% division_decomp
thf(fact_1524_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult
thf(fact_1525_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P5: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ P5 @ ( times_times @ A @ A2 @ B2 ) )
=> ~ ! [X3: A,Y3: A] :
( ( P5
= ( times_times @ A @ X3 @ Y3 ) )
=> ( ( dvd_dvd @ A @ X3 @ A2 )
=> ~ ( dvd_dvd @ A @ Y3 @ B2 ) ) ) ) ) ).
% dvd_productE
thf(fact_1526_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B5: A,A5: A] :
? [K3: A] :
( A5
= ( times_times @ A @ B5 @ K3 ) ) ) ) ) ).
% dvd_def
thf(fact_1527_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A2: A,B2: A,K: A] :
( ( A2
= ( times_times @ A @ B2 @ K ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% dvdI
thf(fact_1528_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ~ ! [K2: A] :
( A2
!= ( times_times @ A @ B2 @ K2 ) ) ) ) ).
% dvdE
thf(fact_1529_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_1530_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_1531_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_1532_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X2: A] : X2 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_1533_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_1534_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_1535_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_1536_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D4 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1537_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D4 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1538_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ).
% zero_le_square
thf(fact_1539_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_1540_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1541_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1542_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_1543_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1544_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_1545_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_1546_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_1547_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1548_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1549_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1550_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1551_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1552_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1553_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_1554_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( times_times @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_1555_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_1556_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_1557_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_1558_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_1559_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_1560_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1561_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_1562_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_1563_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1564_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1565_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1566_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1567_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1568_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1569_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1570_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1571_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1572_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N ) ) ) ) ) ).
% less_1_mult
thf(fact_1573_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A2 )
= A2 ) ) ).
% mult_numeral_1
thf(fact_1574_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
= A2 ) ) ).
% mult_numeral_1_right
thf(fact_1575_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( A2
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( times_times @ A @ A2 @ C2 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1576_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C2 )
= A2 )
= ( B2
= ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1577_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= B2 )
=> ( A2
= ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_1578_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A2 @ C2 ) )
=> ( ( divide_divide @ A @ B2 @ C2 )
= A2 ) ) ) ) ).
% divide_eq_imp
thf(fact_1579_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_1580_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ C2 )
= A2 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_1581_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_1582_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_1583_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D4: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D4 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E2 ) @ C2 )
= D4 ) ) ) ).
% eq_add_iff1
thf(fact_1584_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D4: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D4 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E2 ) @ D4 ) ) ) ) ).
% eq_add_iff2
thf(fact_1585_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_1586_mult__diff__mult,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [X: A,Y: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ Y ) @ ( times_times @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ X @ ( minus_minus @ A @ Y @ B2 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X @ A2 ) @ B2 ) ) ) ) ).
% mult_diff_mult
thf(fact_1587_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_1588_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_1589_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_1590_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_1591_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_1592_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_1593_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_1594_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A2 @ B2 )
= ( times_times @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_1595_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B2 @ A2 )
= ( times_times @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_1596_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,D4: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( dvd_dvd @ A @ D4 @ C2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( divide_divide @ A @ C2 @ D4 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D4 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_1597_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 )
=> ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_1598_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_1599_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_1600_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% div_mult_swap
thf(fact_1601_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).
% dvd_div_mult
thf(fact_1602_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( divide_divide @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_1603_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( power_power @ A @ A2 @ ( plus_plus @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_add
thf(fact_1604_real__minus__mult__self__le,axiom,
! [U2: real,X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( times_times @ real @ U2 @ U2 ) ) @ ( times_times @ real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1605_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ P5 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q3 ) ) ) @ Q3 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_1606_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N ) )
= ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% minus_power_mult_self
thf(fact_1607_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) @ P5 ) ) ) ).
% ceiling_divide_lower
thf(fact_1608_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1609_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_1610_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A2 ) ) ) ).
% ceiling_le
thf(fact_1611_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_1612_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_1613_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_1614_ceiling__divide__eq__div,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: int,B2: int] :
( ( archimedean_ceiling @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A2 ) @ ( ring_1_of_int @ A @ B2 ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 ) ) ) ) ).
% ceiling_divide_eq_div
thf(fact_1615_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_1616_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1617_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1618_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1619_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D4 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1620_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1621_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1622_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D4 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1623_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1624_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1625_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1626_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1627_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1628_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D4 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1629_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D4 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1630_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_1631_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_1632_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_1633_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ A2 ) ) ) ) ).
% mult_left_le
thf(fact_1634_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_1635_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_1636_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_1637_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_1638_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_1639_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1640_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1641_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_1642_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1643_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1644_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_1645_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_1646_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_1647_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1648_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1649_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( numeral_numeral @ A @ W2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1650_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ( numeral_numeral @ A @ W2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1651_ordered__ring__class_Ole__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D4: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D4 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E2 ) @ C2 ) @ D4 ) ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1652_ordered__ring__class_Ole__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D4: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D4 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E2 ) @ D4 ) ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1653_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_1654_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_1655_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_1656_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_1657_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_1658_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A2: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z3 ) )
= A2 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1659_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A2: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z3 ) @ B2 )
= B2 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z3 ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1660_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D4: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D4 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E2 ) @ C2 ) @ D4 ) ) ) ).
% less_add_iff1
thf(fact_1661_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D4: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D4 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E2 ) @ D4 ) ) ) ) ).
% less_add_iff2
thf(fact_1662_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_1663_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_1664_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_1665_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A2: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z3 ) )
= A2 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A2 @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1666_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1667_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1668_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_1669_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_1670_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1671_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_1672_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( C2
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) )
= ( ( times_times @ A @ C2 @ B2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_1673_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= C2 )
= ( ( uminus_uminus @ A @ A2 )
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_1674_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) )
= A2 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B2 )
= ( times_times @ A @ A2 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_1675_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
= ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_1676_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [C5: A] :
( B2
!= ( times_times @ A @ A2 @ C5 ) ) ) ) ) ).
% unit_dvdE
thf(fact_1677_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X2: A] : ( P @ ( times_times @ A @ L @ X2 ) ) )
= ( ? [X2: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X2 @ ( zero_zero @ A ) ) )
& ( P @ X2 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_1678_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A,B2: A,D4: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D4 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ D4 @ C2 ) )
= ( ( times_times @ A @ B2 @ C2 )
= ( times_times @ A @ A2 @ D4 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_1679_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_1680_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_1681_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= C2 )
= ( B2
= ( times_times @ A @ C2 @ A2 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_1682_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_minus
thf(fact_1683_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D4: A,D: A,T2: A] :
( ( dvd_dvd @ A @ D4 @ D )
=> ! [X6: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D4 @ ( plus_plus @ A @ X6 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D4 @ ( plus_plus @ A @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1684_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D4: A,D: A,T2: A] :
( ( dvd_dvd @ A @ D4 @ D )
=> ! [X6: A,K4: A] :
( ( dvd_dvd @ A @ D4 @ ( plus_plus @ A @ X6 @ T2 ) )
= ( dvd_dvd @ A @ D4 @ ( plus_plus @ A @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1685_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= C2 )
= ( A2
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_1686_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( A2
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( times_times @ A @ A2 @ B2 )
= C2 ) ) ) ) ).
% unit_eq_div2
thf(fact_1687_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_1688_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% unit_div_commute
thf(fact_1689_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_1690_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_1691_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ! [Y2: real] :
? [N2: nat] : ( ord_less @ real @ Y2 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1692_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_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_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_1695_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_1696_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1697_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1698_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1699_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1700_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1701_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1702_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1703_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1704_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_1705_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1706_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_1707_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_1708_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_1709_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1710_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1711_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_1712_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_1713_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_1714_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1715_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A2: A,Y: A,U2: A,V2: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ( ord_less_eq @ A @ Y @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U2 @ V2 )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U2 @ X ) @ ( times_times @ A @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1716_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1717_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1718_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_1719_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_1720_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_1721_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_1722_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ B2 ) ) ) ).
% left_add_twice
thf(fact_1723_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_1724_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_1725_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_1726_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_1727_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_1728_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_1729_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_1730_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_1731_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_1732_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_1733_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A2: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z3 ) ) @ B2 )
= B2 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z3 ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1734_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_1735_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A2: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z3 ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z3 ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_1736_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A2: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z3 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z3 ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_1737_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ~ ! [B6: A] :
( A2
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B6 ) ) ) ) ).
% evenE
thf(fact_1738_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [B6: A] :
( ( B6
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B6 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= B6 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B6 )
= A2 )
=> ( ( ( times_times @ A @ A2 @ B6 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A2 )
!= ( times_times @ A @ C2 @ B6 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1739_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_1740_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_1741_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_1742_power2__eq__square,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ A2 @ A2 ) ) ) ).
% power2_eq_square
thf(fact_1743_log__mult,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A2 @ ( times_times @ real @ X @ Y ) )
= ( plus_plus @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_1744_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I4: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I4 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% ceiling_split
thf(fact_1745_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ( archimedean_ceiling @ A @ X )
= A2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_1746_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_1747_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_1748_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_1749_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_1750_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A2: A,Y: A,U2: A,V2: A] :
( ( ord_less @ A @ X @ A2 )
=> ( ( ord_less @ A @ Y @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U2 @ V2 )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U2 @ X ) @ ( times_times @ A @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1751_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_1752_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_1753_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_1754_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_1755_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_1756_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_1757_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_1758_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_1759_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U2: A,V2: A,R3: A,S: A] :
( ( ord_less_eq @ A @ U2 @ V2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R3 )
=> ( ( ord_less_eq @ A @ R3 @ S )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U2 @ ( divide_divide @ A @ ( times_times @ A @ R3 @ ( minus_minus @ A @ V2 @ U2 ) ) @ S ) ) @ V2 ) ) ) ) ) ).
% scaling_mono
thf(fact_1760_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_1761_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_1762_even__two__times__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= A2 ) ) ) ).
% even_two_times_div_two
thf(fact_1763_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P6: A,M3: nat] :
( if @ A
@ ( M3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1764_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A2 )
= ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1765_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_1766_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ! [M4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M4 ) @ X ) @ C2 ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1767_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_1768_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_1769_log__nat__power,axiom,
! [X: real,B2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ B2 @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X ) ) ) ) ).
% log_nat_power
thf(fact_1770_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_1771_log__eq__div__ln__mult__log,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A2 @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( ln_ln @ real @ A2 ) ) @ ( log @ B2 @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_1772_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_1773_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_1774_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ~ ! [B6: A] :
( A2
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B6 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_1775_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_1776_L2__set__mult__ineq__lemma,axiom,
! [A2: real,C2: real,B2: real,D4: real] : ( ord_less_eq @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( times_times @ real @ A2 @ C2 ) ) @ ( times_times @ real @ B2 @ D4 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ real @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ C2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_1777_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_1778_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_1779_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_1780_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_1781_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U2: A,X: A,Y: A] :
( ( ( power_power @ A @ U2 @ ( 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 @ U2 @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_1782_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ord_less @ nat @ N @ M )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_1783_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_1784_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_1785_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_1786_ceiling__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% ceiling_log2_div2
thf(fact_1787_VEBT__internal_Olog__ceil__idem,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ real @ ( archimedean_ceiling @ real @ X ) ) ) ) ) ) ).
% VEBT_internal.log_ceil_idem
thf(fact_1788_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_1789_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_1790_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_1791_tanh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( tanh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% tanh_ln_real
thf(fact_1792_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_1793_ceiling__log__eq__powr__iff,axiom,
! [X: real,B2: real,K: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ B2 @ X ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) ) )
= ( ( ord_less @ real @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ K ) ) @ X )
& ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_1794_floor__log__nat__eq__powr__iff,axiom,
! [B2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N ) )
= ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_1795_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_idempotent
thf(fact_1796_abs__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_abs
thf(fact_1797_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_1798_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_1799_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_1800_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_1801_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_1802_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ A2 ) )
= ( times_times @ A @ A2 @ A2 ) ) ) ).
% abs_mult_self_eq
thf(fact_1803_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_1804_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_add_abs
thf(fact_1805_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_minus_cancel
thf(fact_1806_abs__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_minus
thf(fact_1807_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N @ K ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_1808_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_1809_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_1810_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_1811_abs__dvd__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: A,K: A] :
( ( dvd_dvd @ A @ ( abs_abs @ A @ M ) @ K )
= ( dvd_dvd @ A @ M @ K ) ) ) ).
% abs_dvd_iff
thf(fact_1812_dvd__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: A,K: A] :
( ( dvd_dvd @ A @ M @ ( abs_abs @ A @ K ) )
= ( dvd_dvd @ A @ M @ K ) ) ) ).
% dvd_abs_iff
thf(fact_1813_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_1814_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1815_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( one_one @ nat ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1816_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_1817_powr__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [Z3: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z3 )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_1818_of__int__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int] :
( ( ring_1_of_int @ A @ ( abs_abs @ int @ X ) )
= ( abs_abs @ A @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% of_int_abs
thf(fact_1819_of__int__floor__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) )
= X )
= ( ? [N3: int] :
( X
= ( ring_1_of_int @ A @ N3 ) ) ) ) ) ).
% of_int_floor_cancel
thf(fact_1820_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_1821_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_1822_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times @ num @ M @ one2 )
= M ) ).
% semiring_norm(11)
thf(fact_1823_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times @ num @ one2 @ N )
= N ) ).
% semiring_norm(12)
thf(fact_1824_tanh__real__zero__iff,axiom,
! [X: real] :
( ( ( tanh @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% tanh_real_zero_iff
thf(fact_1825_tanh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( tanh @ real @ X ) @ ( tanh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% tanh_real_less_iff
thf(fact_1826_tanh__real__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X ) @ ( tanh @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% tanh_real_le_iff
thf(fact_1827_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_1828_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% abs_le_self_iff
thf(fact_1829_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_nonneg
thf(fact_1830_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_1831_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_1832_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_1833_abs__power__minus,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N ) )
= ( abs_abs @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% abs_power_minus
thf(fact_1834_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1835_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1836_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1837_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_1838_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K
= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( zero_zero @ nat ) ) )
& ( ( K
!= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( divide_divide @ nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1839_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( dvd_dvd @ nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1840_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_1841_floor__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% floor_numeral
thf(fact_1842_powr__gt__zero,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ A2 ) )
= ( X
!= ( zero_zero @ real ) ) ) ).
% powr_gt_zero
thf(fact_1843_powr__nonneg__iff,axiom,
! [A2: real,X: real] :
( ( ord_less_eq @ real @ ( powr @ real @ A2 @ X ) @ ( zero_zero @ real ) )
= ( A2
= ( zero_zero @ real ) ) ) ).
% powr_nonneg_iff
thf(fact_1844_powr__less__cancel__iff,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
= ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel_iff
thf(fact_1845_num__double,axiom,
! [N: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1846_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_1847_power__mult__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N: num] :
( ( power_power @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% power_mult_numeral
thf(fact_1848_tanh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( tanh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% tanh_real_neg_iff
thf(fact_1849_tanh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% tanh_real_pos_iff
thf(fact_1850_tanh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% tanh_real_nonneg_iff
thf(fact_1851_tanh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% tanh_real_nonpos_iff
thf(fact_1852_zero__le__divide__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_1853_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_0_abs_iff
thf(fact_1854_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% abs_of_nonpos
thf(fact_1855_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1856_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1857_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1858_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1859_powr__eq__one__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ( powr @ real @ A2 @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% powr_eq_one_iff
thf(fact_1860_powr__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( one_one @ real ) )
= X ) ) ).
% powr_one
thf(fact_1861_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr @ real @ X @ ( one_one @ real ) )
= X )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% powr_one_gt_zero_iff
thf(fact_1862_powr__le__cancel__iff,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
= ( ord_less_eq @ real @ A2 @ B2 ) ) ) ).
% powr_le_cancel_iff
thf(fact_1863_artanh__minus__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( artanh @ real @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( artanh @ real @ X ) ) ) ) ).
% artanh_minus_real
thf(fact_1864_numeral__powr__numeral__real,axiom,
! [M: num,N: num] :
( ( powr @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% numeral_powr_numeral_real
thf(fact_1865_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) )
= ( ( A2
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_1866_power2__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_abs
thf(fact_1867_abs__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% abs_power2
thf(fact_1868_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_1869_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_1870_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V2 ) @ X ) ) ) ).
% numeral_le_floor
thf(fact_1871_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_1872_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_1873_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% floor_less_numeral
thf(fact_1874_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_1875_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_1876_floor__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% floor_neg_numeral
thf(fact_1877_floor__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% floor_diff_numeral
thf(fact_1878_floor__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: num,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% floor_numeral_power
thf(fact_1879_log__powr__cancel,axiom,
! [A2: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ ( powr @ real @ A2 @ Y ) )
= Y ) ) ) ).
% log_powr_cancel
thf(fact_1880_powr__log__cancel,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ A2 @ ( log @ A2 @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_1881_floor__divide__eq__div__numeral,axiom,
! [A2: num,B2: num] :
( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_divide_eq_div_numeral
thf(fact_1882_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_1883_power__even__abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: num,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
=> ( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ ( numeral_numeral @ nat @ W2 ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) ) ) ) ).
% power_even_abs_numeral
thf(fact_1884_powr__numeral,axiom,
! [X: real,N: num] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% powr_numeral
thf(fact_1885_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_1886_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_less_floor
thf(fact_1887_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_1888_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_1889_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_1890_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X ) ) ) ).
% neg_numeral_le_floor
thf(fact_1891_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_1892_floor__one__divide__eq__div__numeral,axiom,
! [B2: num] :
( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( divide_divide @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_one_divide_eq_div_numeral
thf(fact_1893_floor__minus__divide__eq__div__numeral,axiom,
! [A2: num,B2: num] :
( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_minus_divide_eq_div_numeral
thf(fact_1894_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_1895_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_less_floor
thf(fact_1896_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_1897_square__powr__half,axiom,
! [X: real] :
( ( powr @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X ) ) ).
% square_powr_half
thf(fact_1898_floor__minus__one__divide__eq__div__numeral,axiom,
! [B2: num] :
( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_minus_one_divide_eq_div_numeral
thf(fact_1899_int__ops_I7_J,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A2 @ B2 ) )
= ( times_times @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_1900_abs__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0_iff
thf(fact_1901_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_self
thf(fact_1902_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% abs_le_D1
thf(fact_1903_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_mult
thf(fact_1904_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_1905_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).
% abs_minus_commute
thf(fact_1906_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_1907_power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ A2 @ N ) )
= ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% power_abs
thf(fact_1908_dvd__if__abs__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [L: A,K: A] :
( ( ( abs_abs @ A @ L )
= ( abs_abs @ A @ K ) )
=> ( dvd_dvd @ A @ L @ K ) ) ) ).
% dvd_if_abs_eq
thf(fact_1909_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1910_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_1911_power__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( power_power @ A @ A2 @ ( times_times @ nat @ M @ N ) )
= ( power_power @ A @ ( power_power @ A @ A2 @ M ) @ N ) ) ) ).
% power_mult
thf(fact_1912_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1913_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1914_left__add__mult__distrib,axiom,
! [I: nat,U2: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U2 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1915_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1916_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1917_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1918_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_1919_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1920_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_1921_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_1922_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_1923_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_1924_div__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( divide_divide @ nat @ M @ ( times_times @ nat @ N @ Q3 ) )
= ( divide_divide @ nat @ ( divide_divide @ nat @ M @ N ) @ Q3 ) ) ).
% div_mult2_eq
thf(fact_1925_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M @ N ) @ K )
= ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1926_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1927_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_1928_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_1929_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A5: nat,B5: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_1930_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_1931_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_1932_take__bit__mult,axiom,
! [N: nat,K: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ K @ L ) ) ) ).
% take_bit_mult
thf(fact_1933_imult__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( times_times @ extended_enat @ M @ N )
= ( zero_zero @ extended_enat ) )
= ( ( M
= ( zero_zero @ extended_enat ) )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% imult_is_0
thf(fact_1934_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_1935_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_zero
thf(fact_1936_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_1937_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_1938_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_pos
thf(fact_1939_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_1940_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_1941_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_1942_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A,B2: A,D4: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ C2 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D4 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D4 ) ) ) ) ) ).
% abs_mult_less
thf(fact_1943_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_1944_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_1945_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_1946_nonzero__abs__divide,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% nonzero_abs_divide
thf(fact_1947_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_minus_self
thf(fact_1948_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_le_iff
thf(fact_1949_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% abs_le_D2
thf(fact_1950_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_leI
thf(fact_1951_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ B2 )
= ( ( ord_less @ A @ A2 @ B2 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_less_iff
thf(fact_1952_powr__non__neg,axiom,
! [A2: real,X: real] :
~ ( ord_less @ real @ ( powr @ real @ A2 @ X ) @ ( zero_zero @ real ) ) ).
% powr_non_neg
thf(fact_1953_powr__less__mono2__neg,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X @ A2 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_1954_powr__mono2,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_mono2
thf(fact_1955_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ Y ) ) ).
% powr_ge_pzero
thf(fact_1956_powr__less__cancel,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel
thf(fact_1957_powr__less__mono,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) ) ) ) ).
% powr_less_mono
thf(fact_1958_powr__mono,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) ) ) ) ).
% powr_mono
thf(fact_1959_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_1960_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_1961_div__mult2__numeral__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,K: num,L: num] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ L ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( times_times @ num @ K @ L ) ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_1962_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1963_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1964_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1965_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1966_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times @ nat @ M @ N ) )
=> ( ( N
= ( one_one @ nat ) )
| ( M
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1967_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_1968_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less @ nat @ M @ ( times_times @ nat @ I @ N ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1969_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1970_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1971_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less @ int @ I @ J )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ord_less @ int @ ( times_times @ int @ K @ I ) @ ( times_times @ int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1972_bezout__lemma__nat,axiom,
! [D4: nat,A2: nat,B2: nat,X: nat,Y: nat] :
( ( dvd_dvd @ nat @ D4 @ A2 )
=> ( ( dvd_dvd @ nat @ D4 @ B2 )
=> ( ( ( ( times_times @ nat @ A2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y ) @ D4 ) )
| ( ( times_times @ nat @ B2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y ) @ D4 ) ) )
=> ? [X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D4 @ A2 )
& ( dvd_dvd @ nat @ D4 @ ( plus_plus @ nat @ A2 @ B2 ) )
& ( ( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ Y3 ) @ D4 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y3 ) @ D4 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1973_bezout__add__nat,axiom,
! [A2: nat,B2: nat] :
? [D6: nat,X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D6 @ A2 )
& ( dvd_dvd @ nat @ D6 @ B2 )
& ( ( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D6 ) )
| ( ( times_times @ nat @ B2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y3 ) @ D6 ) ) ) ) ).
% bezout_add_nat
thf(fact_1974_bezout1__nat,axiom,
! [A2: nat,B2: nat] :
? [D6: nat,X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D6 @ A2 )
& ( dvd_dvd @ nat @ D6 @ B2 )
& ( ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X3 ) @ ( times_times @ nat @ B2 @ Y3 ) )
= D6 )
| ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A2 @ Y3 ) )
= D6 ) ) ) ).
% bezout1_nat
thf(fact_1975_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
= ( ( ( M
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) )
| ( ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
& ( N
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1976_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
=> ( ( M
= ( one_one @ int ) )
| ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1977_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N ) )
=> ( ( K
!= ( zero_zero @ int ) )
=> ( dvd_dvd @ int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_1978_zdvd__mono,axiom,
! [K: int,M: int,T2: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ M @ T2 )
= ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ T2 ) ) ) ) ).
% zdvd_mono
thf(fact_1979_zdvd__period,axiom,
! [A2: int,D4: int,X: int,T2: int,C2: int] :
( ( dvd_dvd @ int @ A2 @ D4 )
=> ( ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ X @ T2 ) )
= ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C2 @ D4 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_1980_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N @ ( times_times @ int @ K @ M ) ) )
= ( dvd_dvd @ int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_1981_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less @ real @ ( tanh @ real @ X ) @ ( one_one @ real ) ) ).
% tanh_real_lt_1
thf(fact_1982_enat__0__less__mult__iff,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( times_times @ extended_enat @ M @ N ) )
= ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ M )
& ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ) ) ).
% enat_0_less_mult_iff
thf(fact_1983_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_1984_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
| ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
| ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_1985_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_1986_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( abs_abs @ A @ B2 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ( B2 = A2 )
| ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_1987_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( abs_abs @ A @ A2 )
= B2 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
& ( ( A2 = B2 )
| ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_1988_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A2 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_1989_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_1990_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% zero_le_power_abs
thf(fact_1991_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% abs_if
thf(fact_1992_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% abs_if_raw
thf(fact_1993_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% abs_of_neg
thf(fact_1994_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D4 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D4 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1995_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_1996_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A2: A,R3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A2 ) ) @ R3 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R3 ) @ X )
& ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A2 @ R3 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1997_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A2: A,R3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A2 ) ) @ R3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ R3 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ A2 @ R3 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1998_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_1999_le__mult__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( nat2 @ ( archim6421214686448440834_floor @ A @ A2 ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ B2 ) ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% le_mult_nat_floor
thf(fact_2000_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_2001_powr__mono2_H,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X @ A2 ) ) ) ) ) ).
% powr_mono2'
thf(fact_2002_powr__less__mono2,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_2003_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_2004_powr__inj,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ( powr @ real @ A2 @ X )
= ( powr @ real @ A2 @ Y ) )
= ( X = Y ) ) ) ) ).
% powr_inj
thf(fact_2005_gr__one__powr,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_2006_powr__le1,axiom,
! [A2: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( one_one @ real ) ) ) ) ) ).
% powr_le1
thf(fact_2007_powr__mono__both,axiom,
! [A2: real,B2: real,X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ B2 ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_2008_ge__one__powr__ge__zero,axiom,
! [X: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( powr @ real @ X @ A2 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_2009_powr__divide,axiom,
! [X: real,Y: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( divide_divide @ real @ X @ Y ) @ A2 )
= ( divide_divide @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_divide
thf(fact_2010_powr__mult,axiom,
! [X: real,Y: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( times_times @ real @ X @ Y ) @ A2 )
= ( times_times @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_mult
thf(fact_2011_floor__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N: nat] :
( ( X
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ X @ N ) )
= ( power_power @ int @ ( archim6421214686448440834_floor @ A @ X ) @ N ) ) ) ) ).
% floor_power
thf(fact_2012_log__base__powr,axiom,
! [A2: real,B2: real,X: real] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( log @ ( powr @ real @ A2 @ B2 ) @ X )
= ( divide_divide @ real @ ( log @ A2 @ X ) @ B2 ) ) ) ).
% log_base_powr
thf(fact_2013_log__powr,axiom,
! [X: real,B2: real,Y: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( log @ B2 @ ( powr @ real @ X @ Y ) )
= ( times_times @ real @ Y @ ( log @ B2 @ X ) ) ) ) ).
% log_powr
thf(fact_2014_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_2015_abs__real__def,axiom,
( ( abs_abs @ real )
= ( ^ [A5: real] : ( if @ real @ ( ord_less @ real @ A5 @ ( zero_zero @ real ) ) @ ( uminus_uminus @ real @ A5 ) @ A5 ) ) ) ).
% abs_real_def
thf(fact_2016_lemma__interval__lt,axiom,
! [A2: real,X: real,B2: real] :
( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [Y2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y2 ) ) @ D6 )
=> ( ( ord_less @ real @ A2 @ Y2 )
& ( ord_less @ real @ Y2 @ B2 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_2017_sin__bound__lemma,axiom,
! [X: real,Y: real,U2: real,V2: real] :
( ( X = Y )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ U2 ) @ V2 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( plus_plus @ real @ X @ U2 ) @ Y ) ) @ V2 ) ) ) ).
% sin_bound_lemma
thf(fact_2018_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_2019_div__less__iff__less__mult,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q3 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q3 ) @ N )
= ( ord_less @ nat @ M @ ( times_times @ nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_2020_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( divide_divide @ nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_2021_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( dvd_dvd @ nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_2022_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( dvd_dvd @ nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_2023_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_2024_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_2025_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_2026_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_2027_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( M
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_2028_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U2 ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_2029_bezout__add__strong__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [D6: nat,X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D6 @ A2 )
& ( dvd_dvd @ nat @ D6 @ B2 )
& ( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D6 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_2030_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
= ( ( M
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_2031_minusinfinity,axiom,
! [D4: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D4 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_2032_plusinfinity,axiom,
! [D4: int,P4: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ! [X3: int,K2: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D4 ) ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [X_1: int] : ( P4 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_2033_zdiv__zmult2__eq,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( divide_divide @ int @ A2 @ ( times_times @ int @ B2 @ C2 ) )
= ( divide_divide @ int @ ( divide_divide @ int @ A2 @ B2 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_2034_tanh__real__gt__neg1,axiom,
! [X: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( tanh @ real @ X ) ) ).
% tanh_real_gt_neg1
thf(fact_2035_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A2 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_2036_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_2037_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_2038_floor__log__eq__powr__iff,axiom,
! [X: real,B2: real,K: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ B2 @ X ) )
= K )
= ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ K ) ) @ X )
& ( ord_less @ real @ X @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_2039_floor__divide__of__nat__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [M: nat,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) ) ).
% floor_divide_of_nat_eq
thf(fact_2040_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_2041_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_2042_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_lessD
thf(fact_2043_powr__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( semiring_1_of_nat @ real @ N ) )
= ( power_power @ real @ X @ N ) ) ) ).
% powr_realpow
thf(fact_2044_powr__less__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X )
= ( ord_less @ real @ Y @ ( log @ B2 @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_2045_less__powr__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( powr @ real @ B2 @ Y ) )
= ( ord_less @ real @ ( log @ B2 @ X ) @ Y ) ) ) ) ).
% less_powr_iff
thf(fact_2046_log__less__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ B2 @ X ) @ Y )
= ( ord_less @ real @ X @ ( powr @ real @ B2 @ Y ) ) ) ) ) ).
% log_less_iff
thf(fact_2047_less__log__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ Y @ ( log @ B2 @ X ) )
= ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_2048_le__nat__floor,axiom,
! [X: nat,A2: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X ) @ A2 )
=> ( ord_less_eq @ nat @ X @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A2 ) ) ) ) ).
% le_nat_floor
thf(fact_2049_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_2050_floor__eq,axiom,
! [N: int,X: real] :
( ( ord_less @ real @ ( ring_1_of_int @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N ) ) ) ).
% floor_eq
thf(fact_2051_real__of__int__floor__add__one__gt,axiom,
! [R3: real] : ( ord_less @ real @ R3 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R3 ) ) @ ( one_one @ real ) ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_2052_real__of__int__floor__add__one__ge,axiom,
! [R3: real] : ( ord_less_eq @ real @ R3 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R3 ) ) @ ( one_one @ real ) ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_2053_real__of__int__floor__gt__diff__one,axiom,
! [R3: real] : ( ord_less @ real @ ( minus_minus @ real @ R3 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R3 ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_2054_real__of__int__floor__ge__diff__one,axiom,
! [R3: real] : ( ord_less_eq @ real @ ( minus_minus @ real @ R3 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R3 ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_2055_lemma__interval,axiom,
! [A2: real,X: real,B2: real] :
( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [Y2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y2 ) ) @ D6 )
=> ( ( ord_less_eq @ real @ A2 @ Y2 )
& ( ord_less_eq @ real @ Y2 @ B2 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_2056_round__diff__minimal,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: A,M: 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 @ M ) ) ) ) ) ).
% round_diff_minimal
thf(fact_2057_power__even__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ ( power_power @ A @ A2 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power_even_eq
thf(fact_2058_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J3 ) )
=> ( P @ I4 ) ) ) ) ) ) ).
% split_div
thf(fact_2059_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_2060_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_2061_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q3 )
=> ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N @ Q3 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_2062_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M3: nat,N3: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N3 @ ( times_times @ nat @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_2063_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_2064_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_2065_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N @ M ) @ M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_2066_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N ) @ M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_2067_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less @ int @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ I ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_2068_dvd__minus__add,axiom,
! [Q3: nat,N: nat,R3: nat,M: nat] :
( ( ord_less_eq @ nat @ Q3 @ N )
=> ( ( ord_less_eq @ nat @ Q3 @ ( times_times @ nat @ R3 @ M ) )
=> ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ Q3 ) )
= ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R3 @ M ) @ Q3 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_2069_unique__quotient__lemma__neg,axiom,
! [B2: int,Q4: int,R4: int,Q3: int,R3: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R3 )
=> ( ( ord_less @ int @ B2 @ R4 )
=> ( ord_less_eq @ int @ Q3 @ Q4 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_2070_unique__quotient__lemma,axiom,
! [B2: int,Q4: int,R4: int,Q3: int,R3: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B2 )
=> ( ( ord_less @ int @ R3 @ B2 )
=> ( ord_less_eq @ int @ Q4 @ Q3 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_2071_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q3: int,R3: int,B8: int,Q4: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R3 )
= ( plus_plus @ int @ ( times_times @ int @ B8 @ Q4 ) @ R4 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B8 @ Q4 ) @ R4 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R3 @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B8 )
=> ( ( ord_less_eq @ int @ B8 @ B2 )
=> ( ord_less_eq @ int @ Q4 @ Q3 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_2072_zdiv__mono2__lemma,axiom,
! [B2: int,Q3: int,R3: int,B8: int,Q4: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R3 )
= ( plus_plus @ int @ ( times_times @ int @ B8 @ Q4 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B8 @ Q4 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B8 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B8 )
=> ( ( ord_less_eq @ int @ B8 @ B2 )
=> ( ord_less_eq @ int @ Q3 @ Q4 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_2073_q__pos__lemma,axiom,
! [B8: int,Q4: int,R4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B8 @ Q4 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B8 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B8 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q4 ) ) ) ) ).
% q_pos_lemma
thf(fact_2074_incr__mult__lemma,axiom,
! [D4: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D4 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( plus_plus @ int @ X6 @ ( times_times @ int @ K @ D4 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_2075_decr__mult__lemma,axiom,
! [D4: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D4 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( minus_minus @ int @ X6 @ ( times_times @ int @ K @ D4 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_2076_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_2077_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A2 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_2078_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I4 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% floor_split
thf(fact_2079_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_2080_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_2081_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_2082_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_2083_abs__square__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( abs_abs @ A @ X )
= ( one_one @ A ) ) ) ) ).
% abs_square_eq_1
thf(fact_2084_power__even__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N )
= ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_even_abs
thf(fact_2085_powr__neg__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ X ) ) ) ).
% powr_neg_one
thf(fact_2086_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_2087_le__log__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y @ ( log @ B2 @ X ) )
= ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_2088_log__le__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ B2 @ X ) @ Y )
= ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ Y ) ) ) ) ) ).
% log_le_iff
thf(fact_2089_le__powr__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ Y ) )
= ( ord_less_eq @ real @ ( log @ B2 @ X ) @ Y ) ) ) ) ).
% le_powr_iff
thf(fact_2090_powr__le__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X )
= ( ord_less_eq @ real @ Y @ ( log @ B2 @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_2091_floor__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N ) ) ) ).
% floor_eq2
thf(fact_2092_floor__divide__real__eq__div,axiom,
! [B2: int,A2: real] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ A2 @ ( ring_1_of_int @ real @ B2 ) ) )
= ( divide_divide @ int @ ( archim6421214686448440834_floor @ real @ A2 ) @ B2 ) ) ) ).
% floor_divide_real_eq_div
thf(fact_2093_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q3 ) ) ) @ Q3 ) @ P5 ) ) ) ).
% floor_divide_lower
thf(fact_2094_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_zdiv
thf(fact_2095_int__div__neg__eq,axiom,
! [A2: int,B2: int,Q3: int,R3: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R3 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_2096_int__div__pos__eq,axiom,
! [A2: int,B2: int,Q3: int,R3: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ R3 @ B2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_2097_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_2098_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_2099_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_2100_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_even
thf(fact_2101_ln__powr__bound,axiom,
! [X: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( divide_divide @ real @ ( powr @ real @ X @ A2 ) @ A2 ) ) ) ) ).
% ln_powr_bound
thf(fact_2102_ln__powr__bound2,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X ) @ A2 ) @ ( times_times @ real @ ( powr @ real @ A2 @ A2 ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_2103_log__add__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ ( log @ B2 @ X ) @ Y )
= ( log @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ Y ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_2104_add__log__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ Y @ ( log @ B2 @ X ) )
= ( log @ B2 @ ( times_times @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_2105_minus__log__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ Y @ ( log @ B2 @ X ) )
= ( log @ B2 @ ( divide_divide @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_2106_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% zero_le_even_power'
thf(fact_2107_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ P5 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) ) ) ) ).
% floor_divide_upper
thf(fact_2108_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X2: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_2109_log__minus__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ ( log @ B2 @ X ) @ Y )
= ( log @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_2110_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ ( zero_zero @ int ) )
=> ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( zero_zero @ int ) )
=> ( P @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_2111_powr__neg__numeral,axiom,
! [X: real,N: num] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_2112_pos__zdiv__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( 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 ) ) @ A2 ) )
= ( divide_divide @ int @ B2 @ A2 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_2113_neg__zdiv__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( 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 ) ) @ A2 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A2 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_2114_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_2115_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_2116_floor__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% floor_log2_div2
thf(fact_2117_powr__int,axiom,
! [X: real,I: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I ) )
= ( power_power @ real @ X @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_2118_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_2119_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_2120_floor__log__nat__eq__if,axiom,
! [B2: nat,N: nat,K: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
=> ( ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_2121_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A > A > $o,X: A] :
( ! [X3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( P @ X3 @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ ( abs_abs @ A @ X ) @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_2122_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_2123_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_2124_arcosh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A )
= ( ^ [X2: A] : ( ln_ln @ A @ ( plus_plus @ A @ X2 @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arcosh_def
thf(fact_2125_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X2: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X2 ) ) @ ( archimedean_ceiling @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ).
% round_altdef
thf(fact_2126_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_2127_arsinh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A )
= ( ^ [X2: A] : ( ln_ln @ A @ ( plus_plus @ A @ X2 @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arsinh_def
thf(fact_2128_arctan__zero__zero,axiom,
( ( arctan @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% arctan_zero_zero
thf(fact_2129_arctan__eq__zero__iff,axiom,
! [X: real] :
( ( ( arctan @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% arctan_eq_zero_iff
thf(fact_2130_zero__less__arctan__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( arctan @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_less_arctan_iff
thf(fact_2131_arctan__less__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( arctan @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% arctan_less_zero_iff
thf(fact_2132_zero__le__arctan__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arctan @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_le_arctan_iff
thf(fact_2133_arctan__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( arctan @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% arctan_le_zero_iff
thf(fact_2134_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd @ int @ X @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_2135_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_2136_zabs__less__one__iff,axiom,
! [Z3: int] :
( ( ord_less @ int @ ( abs_abs @ int @ Z3 ) @ ( one_one @ int ) )
= ( Z3
= ( zero_zero @ int ) ) ) ).
% zabs_less_one_iff
thf(fact_2137_dvd__nat__abs__iff,axiom,
! [N: nat,K: int] :
( ( dvd_dvd @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ).
% dvd_nat_abs_iff
thf(fact_2138_nat__abs__dvd__iff,axiom,
! [K: int,N: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( dvd_dvd @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_abs_dvd_iff
thf(fact_2139_zdvd__antisym__abs,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd @ int @ A2 @ B2 )
=> ( ( dvd_dvd @ int @ B2 @ A2 )
=> ( ( abs_abs @ int @ A2 )
= ( abs_abs @ int @ B2 ) ) ) ) ).
% zdvd_antisym_abs
thf(fact_2140_arctan__monotone,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone
thf(fact_2141_arctan__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% arctan_less_iff
thf(fact_2142_arctan__monotone_H,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone'
thf(fact_2143_arctan__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% arctan_le_iff
thf(fact_2144_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs @ int @ ( times_times @ int @ M @ N ) )
= ( one_one @ int ) )
=> ( ( abs_abs @ int @ M )
= ( one_one @ int ) ) ) ).
% abs_zmult_eq_1
thf(fact_2145_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_2146_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_2147_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_2148_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I4: int] : ( if @ int @ ( ord_less @ int @ I4 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_2149_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_2150_dvd__imp__le__int,axiom,
! [I: int,D4: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ D4 @ I )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ D4 ) @ ( abs_abs @ int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_2151_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_2152_zdvd__mult__cancel1,axiom,
! [M: int,N: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ ( times_times @ int @ M @ N ) @ M )
= ( ( abs_abs @ int @ N )
= ( one_one @ int ) ) ) ) ).
% zdvd_mult_cancel1
thf(fact_2153_div__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_2154_even__abs__add__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ ( abs_abs @ int @ K ) @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_abs_add_iff
thf(fact_2155_even__add__abs__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ ( abs_abs @ int @ L ) ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_add_abs_iff
thf(fact_2156_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_2157_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_2158_nat__abs__int__diff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ B2 @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ A2 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_2159_monoseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( topological_monoseq @ real @ ( power_power @ real @ X ) ) ) ) ).
% monoseq_realpow
thf(fact_2160_incr__lemma,axiom,
! [D4: int,Z3: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( 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 ) ) @ D4 ) ) ) ) ).
% incr_lemma
thf(fact_2161_decr__lemma,axiom,
! [D4: int,X: int,Z3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z3 ) ) @ ( one_one @ int ) ) @ D4 ) ) @ Z3 ) ) ).
% decr_lemma
thf(fact_2162_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_2163_nat0__intermed__int__val,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( F2 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_2164_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_2165_of__real__neg__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W2: num] :
( ( real_Vector_of_real @ A @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ W2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% of_real_neg_numeral
thf(fact_2166_summable__divide__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( F2 @ N3 ) @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_divide_iff
thf(fact_2167_summable__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_cmult_iff
thf(fact_2168_of__real__of__nat__eq,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [N: nat] :
( ( real_Vector_of_real @ A @ ( semiring_1_of_nat @ real @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_real_of_nat_eq
thf(fact_2169_of__real__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,N: nat] :
( ( real_Vector_of_real @ A @ ( power_power @ real @ X @ N ) )
= ( power_power @ A @ ( real_Vector_of_real @ A @ X ) @ N ) ) ) ).
% of_real_power
thf(fact_2170_of__real__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W2: num] :
( ( real_Vector_of_real @ A @ ( numeral_numeral @ real @ W2 ) )
= ( numeral_numeral @ A @ W2 ) ) ) ).
% of_real_numeral
thf(fact_2171_summable__power__series,axiom,
! [F2: nat > real,Z3: real] :
( ! [I2: nat] : ( ord_less_eq @ real @ ( F2 @ I2 ) @ ( one_one @ real ) )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ I2 ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z3 )
=> ( ( ord_less @ real @ Z3 @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I4: nat] : ( times_times @ real @ ( F2 @ I4 ) @ ( power_power @ real @ Z3 @ I4 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_2172_summable__complex__of__real,axiom,
! [F2: nat > real] :
( ( summable @ complex
@ ^ [N3: nat] : ( real_Vector_of_real @ complex @ ( F2 @ N3 ) ) )
= ( summable @ real @ F2 ) ) ).
% summable_complex_of_real
thf(fact_2173_summable__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F2: nat > A] :
( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ).
% summable_single
thf(fact_2174_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N3: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_2175_summable__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ K ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_iff_shift
thf(fact_2176_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_2177_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_2178_summable__const__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: A] :
( ( summable @ A
@ ^ [Uu: nat] : C2 )
= ( C2
= ( zero_zero @ A ) ) ) ) ).
% summable_const_iff
thf(fact_2179_summable__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ C2 ) ) ) ) ).
% summable_mult2
thf(fact_2180_summable__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) ) ) ) ) ).
% summable_mult
thf(fact_2181_summable__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( summable @ A
@ ^ [N3: nat] : ( plus_plus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% summable_add
thf(fact_2182_summable__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( summable @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% summable_diff
thf(fact_2183_summable__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( F2 @ N3 ) @ C2 ) ) ) ) ).
% summable_divide
thf(fact_2184_summable__minus__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( F2 @ N3 ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_minus_iff
thf(fact_2185_summable__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( F2 @ N3 ) ) ) ) ) ).
% summable_minus
thf(fact_2186_summable__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ K ) ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_2187_summable__rabs__cancel,axiom,
! [F2: nat > real] :
( ( summable @ real
@ ^ [N3: nat] : ( abs_abs @ real @ ( F2 @ N3 ) ) )
=> ( summable @ real @ F2 ) ) ).
% summable_rabs_cancel
thf(fact_2188_summable__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_mult_D
thf(fact_2189_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_2190_summable__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X5: nat > real] :
( ( summable @ real @ X5 )
=> ( summable @ A
@ ^ [N3: nat] : ( real_Vector_of_real @ A @ ( X5 @ N3 ) ) ) ) ) ).
% summable_of_real
thf(fact_2191_nonzero__of__real__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [Y: real,X: real] :
( ( Y
!= ( zero_zero @ real ) )
=> ( ( real_Vector_of_real @ A @ ( divide_divide @ real @ X @ Y ) )
= ( divide_divide @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ) ).
% nonzero_of_real_divide
thf(fact_2192_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) ) ) ) ).
% summable_zero_power'
thf(fact_2193_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) ) ) ) ).
% summable_0_powser
thf(fact_2194_summable__powser__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,M: nat,Z3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( plus_plus @ nat @ N3 @ M ) ) @ ( power_power @ A @ Z3 @ N3 ) ) )
= ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_2195_summable__rabs__comparison__test,axiom,
! [F2: nat > real,G: nat > real] :
( ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( F2 @ N2 ) ) @ ( G @ N2 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N3: nat] : ( abs_abs @ real @ ( F2 @ N3 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_2196_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_2197_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X8: nat > A] :
( ! [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) )
| ! [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ M3 ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_2198_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X5: nat > A] :
( ! [M4: nat,N2: nat] :
( ( ord_less_eq @ nat @ M4 @ N2 )
=> ( ord_less_eq @ A @ ( X5 @ N2 ) @ ( X5 @ M4 ) ) )
=> ( topological_monoseq @ A @ X5 ) ) ) ).
% monoI2
thf(fact_2199_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X5: nat > A] :
( ! [M4: nat,N2: nat] :
( ( ord_less_eq @ nat @ M4 @ N2 )
=> ( ord_less_eq @ A @ ( X5 @ M4 ) @ ( X5 @ N2 ) ) )
=> ( topological_monoseq @ A @ X5 ) ) ) ).
% monoI1
thf(fact_2200_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_2201_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_2202_suminf__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% suminf_zero
thf(fact_2203_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_2204_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: nat > A] :
( ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) ) )
= ( F2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_2205_pochhammer__of__nat,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: nat,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( semiring_1_of_nat @ A @ X ) @ N )
= ( semiring_1_of_nat @ A @ ( comm_s3205402744901411588hammer @ nat @ X @ N ) ) ) ) ).
% pochhammer_of_nat
thf(fact_2206_suminf__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X5: nat > real] :
( ( summable @ real @ X5 )
=> ( ( real_Vector_of_real @ A @ ( suminf @ real @ X5 ) )
= ( suminf @ A
@ ^ [N3: nat] : ( real_Vector_of_real @ A @ ( X5 @ N3 ) ) ) ) ) ) ).
% suminf_of_real
thf(fact_2207_suminf__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( G @ N2 ) )
=> ( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) ) ) ) ) ) ).
% suminf_le
thf(fact_2208_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_2209_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_2210_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_2211_suminf__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( ( times_times @ A @ ( suminf @ A @ F2 ) @ C2 )
= ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ C2 ) ) ) ) ) ).
% suminf_mult2
thf(fact_2212_suminf__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) ) )
= ( times_times @ A @ C2 @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_mult
thf(fact_2213_suminf__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ( plus_plus @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) )
= ( suminf @ A
@ ^ [N3: nat] : ( plus_plus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ) ).
% suminf_add
thf(fact_2214_suminf__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) )
= ( suminf @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ) ).
% suminf_diff
thf(fact_2215_suminf__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( F2 @ N3 ) @ C2 ) )
= ( divide_divide @ A @ ( suminf @ A @ F2 ) @ C2 ) ) ) ) ).
% suminf_divide
thf(fact_2216_suminf__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( F2 @ N3 ) ) )
= ( uminus_uminus @ A @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_minus
thf(fact_2217_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ( ( suminf @ A @ F2 )
= ( zero_zero @ A ) )
= ( ! [N3: nat] :
( ( F2 @ N3 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_2218_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_2219_suminf__pos,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_pos
thf(fact_2220_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_2221_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_2222_summable__rabs,axiom,
! [F2: nat > real] :
( ( summable @ real
@ ^ [N3: nat] : ( abs_abs @ real @ ( F2 @ N3 ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( suminf @ real @ F2 ) )
@ ( suminf @ real
@ ^ [N3: nat] : ( abs_abs @ real @ ( F2 @ N3 ) ) ) ) ) ).
% summable_rabs
thf(fact_2223_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_2224_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) )
= ( ? [I4: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_2225_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_2226_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N: nat,K: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N @ K ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_2227_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
= ( zero_zero @ A ) )
= ( ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N )
& ( A2
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_2228_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z3: A,N: nat,M: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z3 @ ( plus_plus @ nat @ N @ M ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ N ) ) @ M ) ) ) ) ).
% pochhammer_product'
thf(fact_2229_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
!= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_2230_binomial__maximum,axiom,
! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% binomial_maximum
thf(fact_2231_binomial__antimono,axiom,
! [K: nat,K5: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K5 )
=> ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K )
=> ( ( ord_less_eq @ nat @ K5 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_2232_binomial__mono,axiom,
! [K: nat,K5: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K5 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K5 ) @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% binomial_mono
thf(fact_2233_binomial__maximum_H,axiom,
! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_2234_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,Z3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z3 @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_2235_binomial__strict__mono,axiom,
! [K: nat,K5: nat,N: nat] :
( ( ord_less @ nat @ K @ K5 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K5 ) @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% binomial_strict_mono
thf(fact_2236_binomial__strict__antimono,axiom,
! [K: nat,K5: nat,N: nat] :
( ( ord_less @ nat @ K @ K5 )
=> ( ( ord_less_eq @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) )
=> ( ( ord_less_eq @ nat @ K5 @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_2237_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R3: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R3 ) @ K ) )
= ( times_times @ A @ R3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R3 ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_2238_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_minus
thf(fact_2239_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ).
% pochhammer_minus'
thf(fact_2240_monoseq__minus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: nat > A] :
( ( topological_monoseq @ A @ A2 )
=> ( topological_monoseq @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( A2 @ N3 ) ) ) ) ) ).
% monoseq_minus
thf(fact_2241_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) )
= ( ord_less_eq @ nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_2242_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_2243_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2244_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= ( zero_zero @ nat ) )
= ( ord_less @ nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_2245_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( times_times @ nat @ K @ ( binomial @ N @ K ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_2246_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_2247_binomial__le__pow2,axiom,
! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_2248_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( binomial @ N @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2249_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus @ nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_2250_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_2251_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_2252_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
= ( times_times @ nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_2253_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_2254_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
@ ^ [N3: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ ( one_one @ real ) ) @ ( suc @ N3 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_2255_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_2256_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N3: nat] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_2257_VEBT__internal_Obit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H: nat,L2: nat,D5: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D5 ) ) @ L2 ) ) ) ).
% VEBT_internal.bit_concat_def
thf(fact_2258_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_2259_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_2260_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_2261_mod__mod__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mod_trivial
thf(fact_2262_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_2263_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_2264_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_2265_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_2266_bits__mod__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_0
thf(fact_2267_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_2268_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% mod_by_0
thf(fact_2269_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_2270_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq @ nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_2271_mod__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_add_self1
thf(fact_2272_mod__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_add_self2
thf(fact_2273_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_2274_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_2275_minus__mod__self2,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% minus_mod_self2
thf(fact_2276_mod__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% mod_minus_minus
thf(fact_2277_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_2278_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_2279_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_2280_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_2281_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_2282_power__Suc0__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% power_Suc0_right
thf(fact_2283_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_2284_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_2285_bits__mod__div__trivial,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_div_trivial
thf(fact_2286_mod__div__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_div_trivial
thf(fact_2287_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self4
thf(fact_2288_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self3
thf(fact_2289_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self2
thf(fact_2290_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self1
thf(fact_2291_dvd__imp__mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( modulo_modulo @ A @ B2 @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% dvd_imp_mod_0
thf(fact_2292_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_2293_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_2294_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_2295_minus__mod__self1,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [B2: A,A2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ B2 @ A2 ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% minus_mod_self1
thf(fact_2296_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times @ nat @ M @ ( suc @ N ) )
= ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_2297_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_2298_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power @ nat @ X @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( X
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_2299_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_2300_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( M
= ( suc @ ( zero_zero @ nat ) ) ) ) ).
% dvd_1_iff_1
thf(fact_2301_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_2302_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_2303_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_2304_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_2305_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% pochhammer_Suc0
thf(fact_2306_of__nat__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ M ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) ) ) ).
% of_nat_Suc
thf(fact_2307_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_2308_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_2309_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) )
= ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
& ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_2310_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ I @ ( suc @ ( minus_minus @ nat @ J @ K ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_2311_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K ) ) @ I )
= ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_2312_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_2313_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_2314_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zless
thf(fact_2315_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_2316_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_2317_bits__one__mod__two__eq__one,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% bits_one_mod_two_eq_one
thf(fact_2318_one__mod__two__eq__one,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% one_mod_two_eq_one
thf(fact_2319_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_2320_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_2321_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_2322_div2__Suc__Suc,axiom,
! [M: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_2323_even__mod__2__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ).
% even_mod_2_iff
thf(fact_2324_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_2325_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_2326_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_2327_zmod__numeral__Bit0,axiom,
! [V2: num,W2: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit0 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W2 ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_2328_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_2329_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_2330_minus__1__mod__2__eq,axiom,
! [A: $tType] :
( ( euclid8789492081693882211th_nat @ A )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% minus_1_mod_2_eq
thf(fact_2331_bits__minus__1__mod__2__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% bits_minus_1_mod_2_eq
thf(fact_2332_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_2333_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_2334_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_2335_take__bit__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A2 )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_0
thf(fact_2336_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_2337_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_2338_of__nat__mod,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) )
= ( modulo_modulo @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mod
thf(fact_2339_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_2340_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_2341_mod__mult__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_mult_eq
thf(fact_2342_mod__mult__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,A7: A,B2: A,B8: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A7 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B8 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A7 @ B8 ) @ C2 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_2343_mod__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_mult_mult2
thf(fact_2344_mult__mod__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( times_times @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( modulo_modulo @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% mult_mod_right
thf(fact_2345_mod__mult__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_mult_left_eq
thf(fact_2346_mod__mult__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_mult_right_eq
thf(fact_2347_mod__add__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_eq
thf(fact_2348_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,A7: A,B2: A,B8: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A7 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B8 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A7 @ B8 ) @ C2 ) ) ) ) ) ).
% mod_add_cong
thf(fact_2349_mod__add__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_left_eq
thf(fact_2350_mod__add__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_right_eq
thf(fact_2351_mod__diff__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_diff_eq
thf(fact_2352_mod__diff__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,A7: A,B2: A,B8: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A7 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B8 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A7 @ B8 ) @ C2 ) ) ) ) ) ).
% mod_diff_cong
thf(fact_2353_mod__diff__left__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_diff_left_eq
thf(fact_2354_mod__diff__right__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_diff_right_eq
thf(fact_2355_mod__minus__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% mod_minus_eq
thf(fact_2356_mod__minus__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A,A7: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( modulo_modulo @ A @ A7 @ B2 ) )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A7 ) @ B2 ) ) ) ) ).
% mod_minus_cong
thf(fact_2357_mod__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% mod_minus_right
thf(fact_2358_power__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ N ) @ B2 )
= ( modulo_modulo @ A @ ( power_power @ A @ A2 @ N ) @ B2 ) ) ) ).
% power_mod
thf(fact_2359_mod__mod__cancel,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ A2 @ C2 ) ) ) ) ).
% mod_mod_cancel
thf(fact_2360_dvd__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [K: A,M: A,N: A] :
( ( dvd_dvd @ A @ K @ M )
=> ( ( dvd_dvd @ A @ K @ N )
=> ( dvd_dvd @ A @ K @ ( modulo_modulo @ A @ M @ N ) ) ) ) ) ).
% dvd_mod
thf(fact_2361_dvd__mod__imp__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_2362_dvd__mod__iff,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ).
% dvd_mod_iff
thf(fact_2363_pi__neq__zero,axiom,
( pi
!= ( zero_zero @ real ) ) ).
% pi_neq_zero
thf(fact_2364_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_2365_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ( ( X
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va: nat] :
( X
!= ( suc @ ( suc @ Va ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_2366_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( ( zero_zero @ nat )
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_2367_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_2368_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_2369_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_2370_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_2371_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_2372_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_2373_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_2374_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_2375_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_2376_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_2377_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_2378_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_2379_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_2380_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_2381_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_2382_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_2383_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_2384_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_2385_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_2386_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( ord_less @ nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_2387_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_2388_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_2389_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less @ nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_2390_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_2391_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_2392_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_2393_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_2394_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_2395_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_2396_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( plus_plus @ nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_2397_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_2398_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A2: nat] :
( ( A3
= ( plus_plus @ nat @ K @ A2 ) )
=> ( ( suc @ A3 )
= ( plus_plus @ nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_2399_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_2400_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_2401_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_2402_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_2403_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq @ nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_2404_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_2405_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq @ nat @ M @ N ) )
= ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_2406_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_2407_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_2408_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_2409_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ ( suc @ K ) @ M )
= ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_2410_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_2411_summable__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_Suc_iff
thf(fact_2412_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,N: nat,I: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N ) @ I )
= ( semiri8178284476397505188at_aux @ A @ Inc @ N @ ( Inc @ I ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_2413_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ A2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2414_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2415_cong__exp__iff__simps_I9_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q3 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_2416_cong__exp__iff__simps_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ one2 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_2417_mod__eq__self__iff__div__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= A2 )
= ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_2418_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
=> ~ ! [D6: A] :
( B2
!= ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ D6 ) ) ) ) ) ).
% mod_eqE
thf(fact_2419_div__add1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_add1_eq
thf(fact_2420_mod__eq__0__iff__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% mod_eq_0_iff_dvd
thf(fact_2421_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B5: A] :
( ( modulo_modulo @ A @ B5 @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_2422_mod__0__imp__dvd,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% mod_0_imp_dvd
thf(fact_2423_mod__eq__dvd__iff,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ C2 @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% mod_eq_dvd_iff
thf(fact_2424_dvd__minus__mod,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A2: A] : ( dvd_dvd @ A @ B2 @ ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% dvd_minus_mod
thf(fact_2425_zmod__le__nonneg__dividend,axiom,
! [M: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ M @ K ) @ M ) ) ).
% zmod_le_nonneg_dividend
thf(fact_2426_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less @ int @ L @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_2427_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( modulo_modulo @ int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_2428_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus1_not_zero
thf(fact_2429_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ K @ ( uminus_uminus @ int @ L ) )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus2_not_zero
thf(fact_2430_zmod__eq__0D,axiom,
! [M: int,D4: int] :
( ( ( modulo_modulo @ int @ M @ D4 )
= ( zero_zero @ int ) )
=> ? [Q5: int] :
( M
= ( times_times @ int @ D4 @ Q5 ) ) ) ).
% zmod_eq_0D
thf(fact_2431_zmod__eq__0__iff,axiom,
! [M: int,D4: int] :
( ( ( modulo_modulo @ int @ M @ D4 )
= ( zero_zero @ int ) )
= ( ? [Q6: int] :
( M
= ( times_times @ int @ D4 @ Q6 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_2432_pi__not__less__zero,axiom,
~ ( ord_less @ real @ pi @ ( zero_zero @ real ) ) ).
% pi_not_less_zero
thf(fact_2433_pi__gt__zero,axiom,
ord_less @ real @ ( zero_zero @ real ) @ pi ).
% pi_gt_zero
thf(fact_2434_pi__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ pi ).
% pi_ge_zero
thf(fact_2435_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N6 )
=> ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ N6 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_2436_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_2437_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_Suc
thf(fact_2438_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ N ) @ A2 ) ) ) ).
% power_Suc2
thf(fact_2439_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N6 )
=> ( ord_less_eq @ A @ ( F2 @ N ) @ ( F2 @ N6 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_2440_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N6 )
=> ( ord_less_eq @ A @ ( F2 @ N6 ) @ ( F2 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_2441_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_2442_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_2443_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_2444_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_2445_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_2446_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( M
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_2447_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_2448_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M @ N ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_2449_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ~ ! [Q5: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M @ Q5 ) ) ) ) ).
% less_natE
thf(fact_2450_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_2451_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_2452_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus @ nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_2453_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_2454_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_2455_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_2456_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_2457_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_2458_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_2459_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_2460_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_2461_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_2462_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N3: nat] : ( ord_less_eq @ nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_2463_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_2464_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_2465_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_2466_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_2467_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_2468_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N ) ) )
= ( minus_minus @ nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_2469_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_2470_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times @ nat @ ( suc @ M ) @ N )
= ( plus_plus @ nat @ N @ ( times_times @ nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_2471_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ ( divide_divide @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_2472_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_2473_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_2474_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_2475_int__cases,axiom,
! [Z3: int] :
( ! [N2: nat] :
( Z3
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( Z3
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_2476_int__of__nat__induct,axiom,
! [P: int > $o,Z3: int] :
( ! [N2: nat] : ( P @ ( semiring_1_of_nat @ int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) )
=> ( P @ Z3 ) ) ) ).
% int_of_nat_induct
thf(fact_2477_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X5: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X5 @ N2 ) @ ( X5 @ ( suc @ N2 ) ) )
=> ( topological_monoseq @ A @ X5 ) ) ) ).
% mono_SucI1
thf(fact_2478_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X5: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X5 @ ( suc @ N2 ) ) @ ( X5 @ N2 ) )
=> ( topological_monoseq @ A @ X5 ) ) ) ).
% mono_SucI2
thf(fact_2479_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X8: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
| ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_2480_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( modulo_modulo @ A @ A2 @ B2 )
= A2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2481_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2482_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q3: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_2483_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_2484_cong__exp__iff__simps_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_2485_cong__exp__iff__simps_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_2486_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_mult1_eq
thf(fact_2487_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B2: A,A2: A] :
( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) )
= A2 ) ) ).
% mult_div_mod_eq
thf(fact_2488_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) )
= A2 ) ) ).
% mod_mult_div_eq
thf(fact_2489_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) )
= A2 ) ) ).
% mod_div_mult_eq
thf(fact_2490_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) )
= A2 ) ) ).
% div_mult_mod_eq
thf(fact_2491_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( A2
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% mod_div_decomp
thf(fact_2492_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A2 @ C2 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_2493_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A2 @ C2 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_2494_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_2495_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_2496_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_2497_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_2498_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_2499_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F3: nat > A > A,A5: nat,B5: nat,Acc: A] : ( if @ A @ ( ord_less @ nat @ B5 @ A5 ) @ Acc @ ( set_fo6178422350223883121st_nat @ A @ F3 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B5 @ ( F3 @ A5 @ Acc ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_2500_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_2501_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ K @ L ) @ ( zero_zero @ int ) ) ) ).
% neg_mod_sign
thf(fact_2502_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_2503_neg__mod__conj,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
& ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).
% neg_mod_conj
thf(fact_2504_pos__mod__conj,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ B2 ) )
& ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ).
% pos_mod_conj
thf(fact_2505_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( ( modulo_modulo @ int @ I @ K )
= I )
= ( ( K
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
& ( ord_less @ int @ I @ K ) )
| ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
& ( ord_less @ int @ K @ I ) ) ) ) ).
% zmod_trivial_iff
thf(fact_2506_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( A2 = B2 ) ) ) ) ) ).
% power_inject_base
thf(fact_2507_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_2508_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_2509_zdiv__mono__strict,axiom,
! [A3: int,B3: int,N: int] :
( ( ord_less @ int @ A3 @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( modulo_modulo @ int @ A3 @ N )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B3 @ N )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A3 @ N ) @ ( divide_divide @ int @ B3 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_2510_zmod__zminus1__eq__if,axiom,
! [A2: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( minus_minus @ int @ B2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_2511_zmod__zminus2__eq__if,axiom,
! [A2: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_2512_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K @ L ) ) @ ( abs_abs @ int @ L ) ) ) ).
% abs_mod_less
thf(fact_2513_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_2514_div__mod__decomp__int,axiom,
! [A3: int,N: int] :
( A3
= ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A3 @ N ) @ N ) @ ( modulo_modulo @ int @ A3 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_2515_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_2516_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_2517_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_2518_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_2519_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_2520_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_2521_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ K @ ( power_power @ nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_2522_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_2523_realpow__pos__nth2,axiom,
! [A2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
& ( ( power_power @ real @ R @ ( suc @ N ) )
= A2 ) ) ) ).
% realpow_pos_nth2
thf(fact_2524_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_2525_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq @ nat @ X3 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq @ nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_2526_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_2527_int__ops_I4_J,axiom,
! [A2: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ A2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( one_one @ int ) ) ) ).
% int_ops(4)
thf(fact_2528_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W3: int,Z7: int] :
? [N3: nat] :
( Z7
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_2529_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_2530_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_2531_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_2532_summable__powser__split__head,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z3 @ N3 ) ) )
= ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) ) ) ) ) ).
% summable_powser_split_head
thf(fact_2533_powser__split__head_I3_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z3 @ N3 ) ) ) ) ) ).
% powser_split_head(3)
thf(fact_2534_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_2535_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_2536_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
= ( plus_plus @ int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_2537_pi__ge__two,axiom,
ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ).
% pi_ge_two
thf(fact_2538_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ) ).
% power_Suc_le_self
thf(fact_2539_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_2540_mod__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= ( modulo_modulo @ int @ ( minus_minus @ int @ K @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_2541_pi__half__neq__two,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% pi_half_neq_two
thf(fact_2542_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( ( L
= ( zero_zero @ int ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_2543_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_2544_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
!= ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_2545_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
!= ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_2546_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_2547_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N )
= ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_2548_real__of__int__div__aux,axiom,
! [X: int,D4: int] :
( ( divide_divide @ real @ ( ring_1_of_int @ real @ X ) @ ( ring_1_of_int @ real @ D4 ) )
= ( plus_plus @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ X @ D4 ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( modulo_modulo @ int @ X @ D4 ) ) @ ( ring_1_of_int @ real @ D4 ) ) ) ) ).
% real_of_int_div_aux
thf(fact_2549_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M3: nat,N3: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ N3
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_2550_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M3: nat,N3: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M3 @ N3 )
| ( N3
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).
% div_if
thf(fact_2551_div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% div_geq
thf(fact_2552_div__nat__eqI,axiom,
! [N: nat,Q3: nat,M: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q3 ) @ M )
=> ( ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q3 ) ) )
=> ( ( divide_divide @ nat @ M @ N )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_2553_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_2554_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A2 @ N ) @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_2555_Suc__nat__number__of__add,axiom,
! [V2: num,N: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V2 @ one2 ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_2556_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_2557_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_2558_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N2: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_2559_Suc__as__int,axiom,
( suc
= ( ^ [A5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( one_one @ int ) ) ) ) ) ).
% Suc_as_int
thf(fact_2560_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_2561_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2562_even__iff__mod__2__eq__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_2563_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_2564_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_2565_take__bit__eq__mod,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A5: A] : ( modulo_modulo @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% take_bit_eq_mod
thf(fact_2566_int__mod__pos__eq,axiom,
! [A2: int,B2: int,Q3: int,R3: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ R3 @ B2 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R3 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_2567_int__mod__neg__eq,axiom,
! [A2: int,B2: int,Q3: int,R3: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R3 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R3 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_2568_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( modulo_modulo @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ N ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_2569_pi__half__neq__zero,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% pi_half_neq_zero
thf(fact_2570_pi__half__less__two,axiom,
ord_less @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% pi_half_less_two
thf(fact_2571_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
= ( minus_minus @ int @ ( minus_minus @ int @ L @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_2572_pi__half__le__two,axiom,
ord_less_eq @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% pi_half_le_two
thf(fact_2573_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_2574_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ~ ! [N2: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ E2 ) ) ) ).
% nat_approx_posE
thf(fact_2575_take__bit__int__def,axiom,
( ( bit_se2584673776208193580ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( modulo_modulo @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% take_bit_int_def
thf(fact_2576_zmod__zmult2__eq,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( modulo_modulo @ int @ A2 @ ( times_times @ int @ B2 @ C2 ) )
= ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_2577_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_2578_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_2579_zdiv__zminus2__eq__if,axiom,
! [B2: int,A2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus2_eq_if
thf(fact_2580_zdiv__zminus1__eq__if,axiom,
! [B2: int,A2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus1_eq_if
thf(fact_2581_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_2582_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_2583_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q6: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q6 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q6 ) ) )
& ( P @ Q6 ) ) ) ) ).
% split_div'
thf(fact_2584_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ nat @ M @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_2585_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_2586_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_2587_bits__stable__imp__add__self,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) ) ) ) ).
% bits_stable_imp_add_self
thf(fact_2588_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_2589_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_2590_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_2591_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_2592_pi__half__gt__zero,axiom,
ord_less @ real @ ( zero_zero @ real ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% pi_half_gt_zero
thf(fact_2593_pi__half__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% pi_half_ge_zero
thf(fact_2594_verit__le__mono__div__int,axiom,
! [A3: int,B3: int,N: int] :
( ( ord_less @ int @ A3 @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B3 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B3 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_2595_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_2596_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_2597_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_2598_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( times_times @ A @ A2 @ ( power_power @ A @ ( power_power @ A @ A2 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_2599_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_2600_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_2601_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_2602_arctan__ubound,axiom,
! [Y: real] : ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_2603_arctan__one,axiom,
( ( arctan @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_one
thf(fact_2604_binomial__less__binomial__Suc,axiom,
! [K: nat,N: nat] :
( ( ord_less @ nat @ K @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_2605_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_2606_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_2607_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_2608_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less @ nat @ I2 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ int @ ( F2 @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ M @ I2 )
& ( ord_less_eq @ nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_2609_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2610_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2611_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) ) )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) ) )
= ( plus_plus @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z3 @ N3 ) ) )
@ Z3 ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_2612_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( suc @ N3 ) ) @ ( power_power @ A @ Z3 @ N3 ) ) )
@ Z3 )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) ) )
@ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_2613_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_2614_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_2615_minus__pi__half__less__zero,axiom,
ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ real ) ).
% minus_pi_half_less_zero
thf(fact_2616_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_2617_arctan__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_bounded
thf(fact_2618_arctan__lbound,axiom,
! [Y: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) ) ).
% arctan_lbound
thf(fact_2619_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X @ M ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2620_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_2621_nat__ivt__aux,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( F2 @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_2622_int__power__div__base,axiom,
! [M: nat,K: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( divide_divide @ int @ ( power_power @ int @ K @ M ) @ K )
= ( power_power @ int @ K @ ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_2623_pos__zmod__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( 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 ) ) @ A2 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A2 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_2624_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_2625_neg__zmod__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( 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 ) ) @ A2 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A2 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_2626_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_2627_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A5: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_2628_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H2: A,Z3: A,K6: real,N: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K6 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z3 @ H2 ) ) @ K6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ N ) @ ( power_power @ A @ Z3 @ N ) ) @ H2 ) @ ( 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 @ K6 @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_2629_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_2630_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A5: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_2631_Suc__if__eq,axiom,
! [A: $tType,F2: nat > A,H2: nat > A,G: A,N: nat] :
( ! [N2: nat] :
( ( F2 @ ( suc @ N2 ) )
= ( H2 @ N2 ) )
=> ( ( ( F2 @ ( zero_zero @ nat ) )
= G )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( ( F2 @ N )
= G ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( F2 @ N )
= ( H2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_2632_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_2633_cos__pi__eq__zero,axiom,
! [M: nat] :
( ( cos @ real @ ( divide_divide @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_eq_zero
thf(fact_2634_bit_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( ( bit_ri4277139882892585799ns_not @ A @ X )
= ( bit_ri4277139882892585799ns_not @ A @ Y ) )
= ( X = Y ) ) ) ).
% bit.compl_eq_compl_iff
thf(fact_2635_bit_Odouble__compl,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= X ) ) ).
% bit.double_compl
thf(fact_2636_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_2637_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_2638_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_2639_bit_Oxor__compl__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ Y ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344971392196577ns_xor @ A @ X @ Y ) ) ) ) ).
% bit.xor_compl_right
thf(fact_2640_bit_Oxor__compl__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ Y )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344971392196577ns_xor @ A @ X @ Y ) ) ) ) ).
% bit.xor_compl_left
thf(fact_2641_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_2642_norm__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% norm_zero
thf(fact_2643_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_2644_norm__numeral,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [W2: num] :
( ( real_V7770717601297561774m_norm @ A @ ( numeral_numeral @ A @ W2 ) )
= ( numeral_numeral @ real @ W2 ) ) ) ).
% norm_numeral
thf(fact_2645_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_2646_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_2647_signed__take__bit__of__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% signed_take_bit_of_minus_1
thf(fact_2648_signed__take__bit__numeral__of__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_numeral_of_1
thf(fact_2649_norm__of__nat,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [N: nat] :
( ( real_V7770717601297561774m_norm @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ real @ N ) ) ) ).
% norm_of_nat
thf(fact_2650_sin__pi,axiom,
( ( sin @ real @ pi )
= ( zero_zero @ real ) ) ).
% sin_pi
thf(fact_2651_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_2652_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_2653_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_2654_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_2655_norm__neg__numeral,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [W2: num] :
( ( real_V7770717601297561774m_norm @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( numeral_numeral @ real @ W2 ) ) ) ).
% norm_neg_numeral
thf(fact_2656_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ K @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_2657_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N @ K ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_2658_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_2659_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ M ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_2660_bit_Oxor__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
= ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% bit.xor_one_left
thf(fact_2661_bit_Oxor__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% bit.xor_one_right
thf(fact_2662_bit_Oxor__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_left
thf(fact_2663_bit_Oxor__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_right
thf(fact_2664_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_2665_not__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% not_nonnegative_int_iff
thf(fact_2666_not__negative__int__iff,axiom,
! [K: int] :
( ( ord_less @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% not_negative_int_iff
thf(fact_2667_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_2668_even__not__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_not_iff
thf(fact_2669_mod2__Suc__Suc,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% mod2_Suc_Suc
thf(fact_2670_norm__mult__numeral1,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [W2: num,A2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ A2 ) )
= ( times_times @ real @ ( numeral_numeral @ real @ W2 ) @ ( real_V7770717601297561774m_norm @ A @ A2 ) ) ) ) ).
% norm_mult_numeral1
thf(fact_2671_norm__mult__numeral2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A,W2: num] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( numeral_numeral @ real @ W2 ) ) ) ) ).
% norm_mult_numeral2
thf(fact_2672_norm__divide__numeral,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A,W2: num] :
( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( numeral_numeral @ real @ W2 ) ) ) ) ).
% norm_divide_numeral
thf(fact_2673_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
!= ( one_one @ nat ) )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N ) ) @ ( numeral_numeral @ nat @ K ) )
= ( one_one @ nat ) ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_2674_sin__npi2,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_npi2
thf(fact_2675_sin__npi,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_npi
thf(fact_2676_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_2677_summable__geometric__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( summable @ A @ ( power_power @ A @ C2 ) )
= ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) ) ) ) ).
% summable_geometric_iff
thf(fact_2678_not__one__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% not_one_eq
thf(fact_2679_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_2680_add__self__mod__2,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ nat ) ) ).
% add_self_mod_2
thf(fact_2681_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_2682_mod2__gr__0,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ nat ) ) ) ).
% mod2_gr_0
thf(fact_2683_signed__take__bit__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_2684_cos__pi__half,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_half
thf(fact_2685_sin__two__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_two_pi
thf(fact_2686_cos__two__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_two_pi
thf(fact_2687_sin__pi__half,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ).
% sin_pi_half
thf(fact_2688_norm__of__real__addn,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: real,B2: num] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( numeral_numeral @ A @ B2 ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X @ ( numeral_numeral @ real @ B2 ) ) ) ) ) ).
% norm_of_real_addn
thf(fact_2689_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_2690_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_2691_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_2692_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_2693_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_2694_sin__cos__squared__add2,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add2
thf(fact_2695_sin__cos__squared__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add
thf(fact_2696_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_2697_sin__of__real__pi__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V7773925162809079976_field @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_of_real_pi_half
thf(fact_2698_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_2699_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_2700_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_2701_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_2702_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_2703_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_2704_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_2705_of__int__not__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int] :
( ( ring_1_of_int @ A @ ( bit_ri4277139882892585799ns_not @ int @ K ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( ring_1_of_int @ A @ K ) ) ) ) ).
% of_int_not_eq
thf(fact_2706_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_2707_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_2708_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M @ N ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_2709_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N ) @ M ) ).
% mod_less_eq_dividend
thf(fact_2710_signed__take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% signed_take_bit_minus
thf(fact_2711_signed__take__bit__mult,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ K @ L ) ) ) ).
% signed_take_bit_mult
thf(fact_2712_signed__take__bit__add,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ K @ L ) ) ) ).
% signed_take_bit_add
thf(fact_2713_take__bit__not__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) ) ) ) ).
% take_bit_not_take_bit
thf(fact_2714_take__bit__not__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ B2 ) ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) ) ) ).
% take_bit_not_iff
thf(fact_2715_signed__take__bit__diff,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ K @ L ) ) ) ).
% signed_take_bit_diff
thf(fact_2716_of__int__not__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ K ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% of_int_not_numeral
thf(fact_2717_sin__zero__abs__cos__one,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ( ( abs_abs @ real @ ( cos @ real @ X ) )
= ( one_one @ real ) ) ) ).
% sin_zero_abs_cos_one
thf(fact_2718_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_2719_sincos__principal__value,axiom,
! [X: real] :
? [Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y3 )
& ( ord_less_eq @ real @ Y3 @ pi )
& ( ( sin @ real @ Y3 )
= ( sin @ real @ X ) )
& ( ( cos @ real @ Y3 )
= ( cos @ real @ X ) ) ) ).
% sincos_principal_value
thf(fact_2720_sin__x__le__x,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( sin @ real @ X ) @ X ) ) ).
% sin_x_le_x
thf(fact_2721_sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( one_one @ real ) ) ).
% sin_le_one
thf(fact_2722_not__diff__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ).
% not_diff_distrib
thf(fact_2723_not__add__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ).
% not_add_distrib
thf(fact_2724_cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( one_one @ real ) ) ).
% cos_le_one
thf(fact_2725_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo @ nat @ M @ N ) )
= N )
=> ( ( modulo_modulo @ nat @ ( suc @ M ) @ N )
= ( zero_zero @ nat ) ) )
& ( ( ( suc @ ( modulo_modulo @ nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo @ nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo @ nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_2726_mod__induct,axiom,
! [P: nat > $o,N: nat,P5: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less @ nat @ N @ P5 )
=> ( ( ord_less @ nat @ M @ P5 )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ P5 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N2 ) @ P5 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_2727_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( modulo_modulo @ nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_2728_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M4: nat] : ( P @ M4 @ ( zero_zero @ nat ) )
=> ( ! [M4: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ N2 @ ( modulo_modulo @ nat @ M4 @ N2 ) )
=> ( P @ M4 @ N2 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_2729_norm__not__less__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
~ ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( zero_zero @ real ) ) ) ).
% norm_not_less_zero
thf(fact_2730_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_2731_norm__ge__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ).
% norm_ge_zero
thf(fact_2732_abs__sin__x__le__abs__x,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X ) ) @ ( abs_abs @ real @ X ) ) ).
% abs_sin_x_le_abs_x
thf(fact_2733_norm__power,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,N: nat] :
( ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X @ N ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ N ) ) ) ).
% norm_power
thf(fact_2734_mod__eq__0D,axiom,
! [M: nat,D4: nat] :
( ( ( modulo_modulo @ nat @ M @ D4 )
= ( zero_zero @ nat ) )
=> ? [Q5: nat] :
( M
= ( times_times @ nat @ D4 @ Q5 ) ) ) ).
% mod_eq_0D
thf(fact_2735_mod__geq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ).
% mod_geq
thf(fact_2736_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M3: nat,N3: nat] : ( if @ nat @ ( ord_less @ nat @ M3 @ N3 ) @ M3 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M3 @ N3 ) @ N3 ) ) ) ) ).
% mod_if
thf(fact_2737_le__mod__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( modulo_modulo @ nat @ M @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_2738_complex__mod__minus__le__complex__mod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ X ) ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_2739_complex__mod__triangle__ineq2,axiom,
! [B2: complex,A2: complex] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ B2 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ B2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ A2 ) ) ).
% complex_mod_triangle_ineq2
thf(fact_2740_cos__arctan__not__zero,axiom,
! [X: real] :
( ( cos @ real @ ( arctan @ X ) )
!= ( zero_zero @ real ) ) ).
% cos_arctan_not_zero
thf(fact_2741_zmod__int,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ A2 @ B2 ) )
= ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zmod_int
thf(fact_2742_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_2743_sin__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_squared_eq
thf(fact_2744_cos__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_squared_eq
thf(fact_2745_summable__norm__cancel,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A] :
( ( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) )
=> ( summable @ A @ F2 ) ) ) ).
% summable_norm_cancel
thf(fact_2746_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 )
= ( bit_ri4674362597316999326ke_bit @ A @ N @ B2 ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ B2 ) ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_2747_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A2 ) ) ) ).
% signed_take_bit_take_bit
thf(fact_2748_sin__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ pi )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero
thf(fact_2749_sin__x__ge__neg__x,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ X ) @ ( sin @ real @ X ) ) ) ).
% sin_x_ge_neg_x
thf(fact_2750_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A5: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_2751_sin__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_ge_zero
thf(fact_2752_nonzero__norm__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ) ).
% nonzero_norm_divide
thf(fact_2753_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_2754_not__eq__complement,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A )
= ( ^ [A5: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% not_eq_complement
thf(fact_2755_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_2756_sin__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( sin @ real @ X ) ) ).
% sin_ge_minus_one
thf(fact_2757_cos__inj__pi,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ( cos @ real @ X )
= ( cos @ real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% cos_inj_pi
thf(fact_2758_cos__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_le_eq
thf(fact_2759_cos__monotone__0__pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi_le
thf(fact_2760_norm__mult__less,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A,R3: real,Y: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R3 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y ) ) @ ( times_times @ real @ R3 @ S ) ) ) ) ) ).
% norm_mult_less
thf(fact_2761_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_2762_cos__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( cos @ real @ X ) ) ).
% cos_ge_minus_one
thf(fact_2763_not__int__def,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K3: int] : ( minus_minus @ int @ ( uminus_uminus @ int @ K3 ) @ ( one_one @ int ) ) ) ) ).
% not_int_def
thf(fact_2764_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_2765_abs__sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X ) ) @ ( one_one @ real ) ) ).
% abs_sin_le_one
thf(fact_2766_norm__triangle__lt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E2: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ E2 ) ) ) ).
% norm_triangle_lt
thf(fact_2767_norm__add__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,R3: real,Y: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R3 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ real @ R3 @ S ) ) ) ) ) ).
% norm_add_less
thf(fact_2768_norm__triangle__mono,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,R3: real,B2: A,S: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ R3 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ S )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ R3 @ S ) ) ) ) ) ).
% norm_triangle_mono
thf(fact_2769_norm__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_triangle_ineq
thf(fact_2770_norm__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E2: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ E2 ) ) ) ).
% norm_triangle_le
thf(fact_2771_norm__add__leD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C2: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ C2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ C2 ) ) ) ) ).
% norm_add_leD
thf(fact_2772_div__less__mono,axiom,
! [A3: nat,B3: nat,N: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A3 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B3 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A3 @ N ) @ ( divide_divide @ nat @ B3 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_2773_norm__diff__triangle__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E1: real,Z3: A,E22: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E1 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ Z3 ) ) @ E22 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Z3 ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_2774_norm__power__ineq,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A,N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X @ N ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ N ) ) ) ).
% norm_power_ineq
thf(fact_2775_abs__cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( cos @ real @ X ) ) @ ( one_one @ real ) ) ).
% abs_cos_le_one
thf(fact_2776_norm__triangle__le__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E2: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E2 ) ) ) ).
% norm_triangle_le_diff
thf(fact_2777_norm__diff__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E1: real,Z3: A,E22: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E1 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ Z3 ) ) @ E22 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Z3 ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_2778_norm__triangle__ineq4,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ).
% norm_triangle_ineq4
thf(fact_2779_norm__triangle__sub,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) ) ) ) ).
% norm_triangle_sub
thf(fact_2780_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ N ) )
= ( ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_2781_mod__eq__nat1E,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q3 )
= ( modulo_modulo @ nat @ N @ Q3 ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ~ ! [S3: nat] :
( M
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q3 @ S3 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_2782_mod__eq__nat2E,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q3 )
= ( modulo_modulo @ nat @ N @ Q3 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ~ ! [S3: nat] :
( N
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q3 @ S3 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_2783_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 )
=> ? [Q5: nat] :
( X
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q5 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_2784_norm__diff__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% norm_diff_ineq
thf(fact_2785_norm__triangle__ineq2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% norm_triangle_ineq2
thf(fact_2786_div__mod__decomp,axiom,
! [A3: nat,N: nat] :
( A3
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A3 @ N ) @ N ) @ ( modulo_modulo @ nat @ A3 @ N ) ) ) ).
% div_mod_decomp
thf(fact_2787_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N @ Q3 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N ) @ Q3 ) ) @ ( modulo_modulo @ nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_2788_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_2789_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_2790_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_2791_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_2792_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_2793_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_2794_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_2795_modulo__nat__def,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M3: nat,N3: nat] : ( minus_minus @ nat @ M3 @ ( times_times @ nat @ ( divide_divide @ nat @ M3 @ N3 ) @ N3 ) ) ) ) ).
% modulo_nat_def
thf(fact_2796_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q3: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( ( modulo_modulo @ nat @ M @ Q3 )
= ( modulo_modulo @ nat @ N @ Q3 ) )
= ( dvd_dvd @ nat @ Q3 @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_2797_sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sin @ A )
= ( ^ [X2: A] : ( cos @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) ) ) ).
% sin_cos_eq
thf(fact_2798_cos__sin__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cos @ A )
= ( ^ [X2: A] : ( sin @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) ) ) ).
% cos_sin_eq
thf(fact_2799_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_2800_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( G @ N2 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_2801_minus__sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( sin @ A @ X ) )
= ( cos @ A @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minus_sin_cos_eq
thf(fact_2802_nat__mod__as__int,axiom,
( ( modulo_modulo @ nat )
= ( ^ [A5: nat,B5: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_2803_not__int__div__2,axiom,
! [K: int] :
( ( divide_divide @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% not_int_div_2
thf(fact_2804_even__not__iff__int,axiom,
! [K: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ K ) )
= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).
% even_not_iff_int
thf(fact_2805_cos__two__neq__zero,axiom,
( ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% cos_two_neq_zero
thf(fact_2806_cos__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_2807_cos__monotone__0__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_2808_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_2809_sin__eq__0__pi,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( ord_less @ real @ X @ pi )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% sin_eq_0_pi
thf(fact_2810_norm__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D4 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ D4 ) ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_2811_even__even__mod__4__iff,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_2812_sin__zero__pi__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ pi )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% sin_zero_pi_iff
thf(fact_2813_cos__monotone__minus__pi__0_H,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_2814_norm__less__p1,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ ( real_V7770717601297561774m_norm @ A @ X ) ) @ ( one_one @ A ) ) ) ) ) ).
% norm_less_p1
thf(fact_2815_field__char__0__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_2816_split__mod,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( modulo_modulo @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ M ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_2817_sin__zero__iff__int2,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I4: int] :
( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I4 ) @ pi ) ) ) ) ).
% sin_zero_iff_int2
thf(fact_2818_mod__nat__eqI,axiom,
! [R3: nat,N: nat,M: nat] :
( ( ord_less @ nat @ R3 @ N )
=> ( ( ord_less_eq @ nat @ R3 @ M )
=> ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M @ R3 ) )
=> ( ( modulo_modulo @ nat @ M @ N )
= R3 ) ) ) ) ).
% mod_nat_eqI
thf(fact_2819_norm__triangle__ineq3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% norm_triangle_ineq3
thf(fact_2820_real__of__nat__div__aux,axiom,
! [X: nat,D4: nat] :
( ( divide_divide @ real @ ( semiring_1_of_nat @ real @ X ) @ ( semiring_1_of_nat @ real @ D4 ) )
= ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ X @ D4 ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ ( modulo_modulo @ nat @ X @ D4 ) ) @ ( semiring_1_of_nat @ real @ D4 ) ) ) ) ).
% real_of_nat_div_aux
thf(fact_2821_sincos__total__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ pi )
& ( X
= ( cos @ real @ T4 ) )
& ( Y
= ( sin @ real @ T4 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_2822_summable__comparison__test_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G: nat > real,N4: nat,F2: nat > A] :
( ( summable @ real @ G )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( G @ N2 ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test'
thf(fact_2823_summable__comparison__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( G @ N2 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test
thf(fact_2824_nat__mod__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( modulo_modulo @ int @ X @ Y ) )
= ( modulo_modulo @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_2825_sin__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_2826_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_2827_mod__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( modulo_modulo @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% mod_abs_eq_div_nat
thf(fact_2828_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,X: A,Z3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) ) ) ) ) ) ).
% powser_inside
thf(fact_2829_summable__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ C2 ) ) ) ) ).
% summable_geometric
thf(fact_2830_complete__algebra__summable__geometric,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ X ) ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_2831_cos__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ real @ ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_2832_sin__gt__zero__02,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero_02
thf(fact_2833_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_2834_cos__is__zero,axiom,
? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X3 )
= ( zero_zero @ real ) )
& ! [Y2: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
& ( ord_less_eq @ real @ Y2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ Y2 )
= ( zero_zero @ real ) ) )
=> ( Y2 = X3 ) ) ) ).
% cos_is_zero
thf(fact_2835_cos__two__le__zero,axiom,
ord_less_eq @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_le_zero
thf(fact_2836_summable__norm,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A] :
( ( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( suminf @ A @ F2 ) )
@ ( suminf @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) ) ) ) ) ).
% summable_norm
thf(fact_2837_cos__monotone__minus__pi__0,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_2838_cos__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ pi )
& ( ( cos @ real @ X3 )
= Y )
& ! [Y2: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
& ( ord_less_eq @ real @ Y2 @ pi )
& ( ( cos @ real @ Y2 )
= Y ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% cos_total
thf(fact_2839_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N ) ) @ M )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_2840_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: A > B] :
( ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X5 @ N3 ) ) @ K7 ) ) )
= ( ? [N7: nat] :
! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X5 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N7 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_2841_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: A > B] :
( ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X5 @ N3 ) ) @ K7 ) ) )
= ( ? [N7: nat] :
! [N3: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X5 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N7 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_2842_signed__take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_2843_norm__of__real__diff,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [B2: real,A2: real] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( real_Vector_of_real @ A @ B2 ) @ ( real_Vector_of_real @ A @ A2 ) ) ) @ ( abs_abs @ real @ ( minus_minus @ real @ B2 @ A2 ) ) ) ) ).
% norm_of_real_diff
thf(fact_2844_sincos__total__pi__half,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( X
= ( cos @ real @ T4 ) )
& ( Y
= ( sin @ real @ T4 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_2845_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 ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
& ( X
= ( cos @ real @ T4 ) )
& ( Y
= ( sin @ real @ T4 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_2846_take__bit__nat__def,axiom,
( ( bit_se2584673776208193580ke_bit @ nat )
= ( ^ [N3: nat,M3: nat] : ( modulo_modulo @ nat @ M3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% take_bit_nat_def
thf(fact_2847_even__signed__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ M @ A2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ).
% even_signed_take_bit_iff
thf(fact_2848_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,X: A,Z3: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_2849_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 ) )
=> ~ ! [T4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
=> ( ( ord_less @ real @ T4 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ( X
= ( cos @ real @ T4 ) )
=> ( Y
!= ( sin @ real @ T4 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_2850_square__norm__one,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ X )
= ( one_one @ real ) ) ) ) ).
% square_norm_one
thf(fact_2851_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_2852_verit__le__mono__div,axiom,
! [A3: nat,B3: nat,N: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A3 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B3 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B3 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_2853_norm__power__diff,axiom,
! [A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z3: A,W2: A,M: 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 @ M ) @ ( power_power @ A @ W2 @ M ) ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Z3 @ W2 ) ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_2854_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_2855_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_2856_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_2857_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_2858_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_2859_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_2860_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_2861_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R3: real,F2: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ( ( summable @ A @ F2 )
=> ? [N8: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ N8 @ N9 )
=> ( ord_less @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( suminf @ A
@ ^ [I4: nat] : ( F2 @ ( plus_plus @ nat @ I4 @ N9 ) ) ) )
@ R3 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_2862_Abel__lemma,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R3: real,R0: real,A2: nat > A,M7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R3 )
=> ( ( ord_less @ real @ R3 @ R0 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N2 ) ) @ ( power_power @ real @ R0 @ N2 ) ) @ M7 )
=> ( summable @ real
@ ^ [N3: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N3 ) ) @ ( power_power @ real @ R3 @ N3 ) ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_2863_sin__gt__zero2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero2
thf(fact_2864_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_2865_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_2866_cos__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_gt_zero
thf(fact_2867_sin__inj__pi,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( sin @ real @ X )
= ( sin @ real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_2868_sin__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_2869_sin__monotone__2pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_2870_cos__one__2pi__int,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( one_one @ real ) )
= ( ? [X2: int] :
( X
= ( times_times @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_2871_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C2: real,N4: nat,F2: nat > A] :
( ( ord_less @ real @ C2 @ ( one_one @ real ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ ( suc @ N2 ) ) ) @ ( times_times @ real @ C2 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_ratio_test
thf(fact_2872_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_2873_signed__take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
=> ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_2874_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_2875_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K )
= ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_2876_suminf__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
=> ( ( suminf @ A @ ( power_power @ A @ C2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ).
% suminf_geometric
thf(fact_2877_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_2878_sin__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_less_zero
thf(fact_2879_sin__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_2880_sin__monotone__2pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_2881_sin__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X3 )
= Y )
& ! [Y2: real] :
( ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
& ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ Y2 )
= Y ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% sin_total
thf(fact_2882_cos__gt__zero__pi,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_2883_cos__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_ge_zero
thf(fact_2884_cos__one__2pi,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( one_one @ real ) )
= ( ? [X2: nat] :
( X
= ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
| ? [X2: nat] :
( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ) ).
% cos_one_2pi
thf(fact_2885_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_2886_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M3: nat,N3: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ N3
@ ( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ M3
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_2887_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_2888_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_2889_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_2890_sin__zero__iff__int,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I4: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I4 )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_2891_cos__zero__iff__int,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I4: int] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I4 )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I4 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_2892_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_2893_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_2894_sin__zero__iff,axiom,
! [X: real] :
( ( ( 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 ) ) ) ) ) )
| ? [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_2895_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
=> ? [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_2896_cos__zero__iff,axiom,
! [X: real] :
( ( ( 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 ) ) ) ) ) )
| ? [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_2897_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_2898_triangle__def,axiom,
( nat_triangle
= ( ^ [N3: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_2899_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
@ ^ [N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) @ ( power_power @ A @ Z3 @ N3 ) )
@ ( 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_2900_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_2901_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_2902_complex__unimodular__polar,axiom,
! [Z3: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z3 )
= ( one_one @ real ) )
=> ~ ! [T4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
=> ( ( ord_less @ real @ T4 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( Z3
!= ( complex2 @ ( cos @ real @ T4 ) @ ( sin @ real @ T4 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_2903_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_2904_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_2905_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_2906_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_2907_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one2 ) ).
% semiring_norm(86)
thf(fact_2908_semiring__norm_I84_J,axiom,
! [N: num] :
( one2
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_2909_tan__pi,axiom,
( ( tan @ real @ pi )
= ( zero_zero @ real ) ) ).
% tan_pi
thf(fact_2910_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_2911_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_2912_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_2913_sums__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( sums @ A
@ ^ [N3: nat] : ( zero_zero @ A )
@ ( zero_zero @ A ) ) ) ).
% sums_zero
thf(fact_2914_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_2915_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_2916_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_2917_semiring__norm_I14_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times @ num @ M @ ( bit1 @ N ) ) ) ) ).
% semiring_norm(14)
thf(fact_2918_semiring__norm_I15_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times @ num @ ( bit1 @ M ) @ N ) ) ) ).
% semiring_norm(15)
thf(fact_2919_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_2920_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_2921_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% semiring_norm(70)
thf(fact_2922_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_2923_zdiv__numeral__Bit1,axiom,
! [V2: num,W2: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit1 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W2 ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_2924_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ one2 ) ) ) ).
% semiring_norm(10)
thf(fact_2925_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
= ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).
% semiring_norm(8)
thf(fact_2926_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_2927_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% semiring_norm(4)
thf(fact_2928_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_2929_tan__npi,axiom,
! [N: nat] :
( ( tan @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% tan_npi
thf(fact_2930_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_2931_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_2932_semiring__norm_I16_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_2933_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_2934_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_2935_xor__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ X ) ) ) ) ).
% xor_numerals(8)
thf(fact_2936_xor__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% xor_numerals(5)
thf(fact_2937_xor__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) ) ).
% xor_numerals(2)
thf(fact_2938_xor__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% xor_numerals(1)
thf(fact_2939_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A,X: A] :
( ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) )
@ X )
= ( ( A2 @ ( zero_zero @ nat ) )
= X ) ) ) ).
% powser_sums_zero_iff
thf(fact_2940_div__Suc__eq__div__add3,axiom,
! [M: nat,N: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( divide_divide @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_2941_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V2: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_2942_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_2943_mod__Suc__eq__mod__add3,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( modulo_modulo @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_2944_Suc__mod__eq__add3__mod__numeral,axiom,
! [M: nat,V2: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_2945_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_2946_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_2947_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_2948_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_2949_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_2950_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_2951_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_2952_zmod__numeral__Bit1,axiom,
! [V2: num,W2: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W2 ) ) ) @ ( one_one @ int ) ) ) ).
% zmod_numeral_Bit1
thf(fact_2953_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_2954_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_2955_sums__0,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A] :
( ! [N2: nat] :
( ( F2 @ N2 )
= ( zero_zero @ A ) )
=> ( sums @ A @ F2 @ ( zero_zero @ A ) ) ) ) ).
% sums_0
thf(fact_2956_sums__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,G: nat > A,S: A,T2: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( G @ N2 ) )
=> ( ( sums @ A @ F2 @ S )
=> ( ( sums @ A @ G @ T2 )
=> ( ord_less_eq @ A @ S @ T2 ) ) ) ) ) ).
% sums_le
thf(fact_2957_sums__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X5: nat > real,A2: real] :
( ( sums @ real @ X5 @ A2 )
=> ( sums @ A
@ ^ [N3: nat] : ( real_Vector_of_real @ A @ ( X5 @ N3 ) )
@ ( real_Vector_of_real @ A @ A2 ) ) ) ) ).
% sums_of_real
thf(fact_2958_sums__of__real__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > real,C2: real] :
( ( sums @ A
@ ^ [N3: nat] : ( real_Vector_of_real @ A @ ( F2 @ N3 ) )
@ ( real_Vector_of_real @ A @ C2 ) )
= ( sums @ real @ F2 @ C2 ) ) ) ).
% sums_of_real_iff
thf(fact_2959_Complex__eq__0,axiom,
! [A2: real,B2: real] :
( ( ( complex2 @ A2 @ B2 )
= ( zero_zero @ complex ) )
= ( ( A2
= ( zero_zero @ real ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_0
thf(fact_2960_zero__complex_Ocode,axiom,
( ( zero_zero @ complex )
= ( complex2 @ ( zero_zero @ real ) @ ( zero_zero @ real ) ) ) ).
% zero_complex.code
thf(fact_2961_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_2962_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one2
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_2963_sums__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F2: nat > A] :
( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) )
@ ( F2 @ I ) ) ) ).
% sums_single
thf(fact_2964_sums__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,A2: A,C2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ C2 )
@ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% sums_mult2
thf(fact_2965_sums__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: nat > A,A2: A,C2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) )
@ ( times_times @ A @ C2 @ A2 ) ) ) ) ).
% sums_mult
thf(fact_2966_sums__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,A2: A,G: nat > A,B2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( ( sums @ A @ G @ B2 )
=> ( sums @ A
@ ^ [N3: nat] : ( plus_plus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% sums_add
thf(fact_2967_sums__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,A2: A,G: nat > A,B2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( ( sums @ A @ G @ B2 )
=> ( sums @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% sums_diff
thf(fact_2968_sums__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,A2: A,C2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( sums @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( F2 @ N3 ) @ C2 )
@ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ).
% sums_divide
thf(fact_2969_sums__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,A2: A] :
( ( sums @ A @ F2 @ A2 )
=> ( sums @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( F2 @ N3 ) )
@ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% sums_minus
thf(fact_2970_one__complex_Ocode,axiom,
( ( one_one @ complex )
= ( complex2 @ ( one_one @ real ) @ ( zero_zero @ real ) ) ) ).
% one_complex.code
thf(fact_2971_Complex__eq__1,axiom,
! [A2: real,B2: real] :
( ( ( complex2 @ A2 @ B2 )
= ( one_one @ complex ) )
= ( ( A2
= ( one_one @ real ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_1
thf(fact_2972_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one2 )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_2973_Complex__eq__numeral,axiom,
! [A2: real,B2: real,W2: num] :
( ( ( complex2 @ A2 @ B2 )
= ( numeral_numeral @ complex @ W2 ) )
= ( ( A2
= ( numeral_numeral @ real @ W2 ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_numeral
thf(fact_2974_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) )
@ ( times_times @ A @ C2 @ D4 ) )
= ( sums @ A @ F2 @ D4 ) ) ) ) ).
% sums_mult_iff
thf(fact_2975_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ N3 ) @ C2 )
@ ( times_times @ A @ D4 @ C2 ) )
= ( sums @ A @ F2 @ D4 ) ) ) ) ).
% sums_mult2_iff
thf(fact_2976_complex__of__real__def,axiom,
( ( real_Vector_of_real @ complex )
= ( ^ [R5: real] : ( complex2 @ R5 @ ( zero_zero @ real ) ) ) ) ).
% complex_of_real_def
thf(fact_2977_complex__of__real__code,axiom,
( ( real_Vector_of_real @ complex )
= ( ^ [X2: real] : ( complex2 @ X2 @ ( zero_zero @ real ) ) ) ) ).
% complex_of_real_code
thf(fact_2978_complex__eq__cancel__iff2,axiom,
! [X: real,Y: real,Xa: real] :
( ( ( complex2 @ X @ Y )
= ( real_Vector_of_real @ complex @ Xa ) )
= ( ( X = Xa )
& ( Y
= ( zero_zero @ real ) ) ) ) ).
% complex_eq_cancel_iff2
thf(fact_2979_sums__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A,A2: A] :
( ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( F2 @ N3 ) )
@ A2 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( sums @ A @ F2 @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ).
% sums_mult_D
thf(fact_2980_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ S )
=> ( sums @ A @ F2 @ S ) ) ) ) ).
% sums_Suc_imp
thf(fact_2981_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A] :
( ( sums @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ S )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_2982_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,L: A] :
( ( sums @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ L )
=> ( sums @ A @ F2 @ ( plus_plus @ A @ L @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_2983_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N: nat,F2: nat > A,S: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( F2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I4: nat] : ( F2 @ ( plus_plus @ nat @ I4 @ N ) )
@ S )
= ( sums @ A @ F2 @ S ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_2984_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_2985_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_2986_Complex__eq__neg__1,axiom,
! [A2: real,B2: real] :
( ( ( complex2 @ A2 @ B2 )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) )
= ( ( A2
= ( uminus_uminus @ real @ ( one_one @ real ) ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_neg_1
thf(fact_2987_Complex__eq__neg__numeral,axiom,
! [A2: real,B2: real,W2: num] :
( ( ( complex2 @ A2 @ B2 )
= ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W2 ) ) )
= ( ( A2
= ( uminus_uminus @ real @ ( numeral_numeral @ real @ W2 ) ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_neg_numeral
thf(fact_2988_cong__exp__iff__simps_I13_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q3 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_2989_cong__exp__iff__simps_I12_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_2990_cong__exp__iff__simps_I10_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_2991_power__minus__Bit1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_2992_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_2993_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_2994_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M: nat,Z3: A] :
( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( if @ A @ ( N3 = M ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z3 @ N3 ) )
@ ( power_power @ A @ Z3 @ M ) ) ) ).
% powser_sums_if
thf(fact_2995_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A] :
( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N3 ) )
@ ( A2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_2996_numeral__Bit1__div__2,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% numeral_Bit1_div_2
thf(fact_2997_odd__numeral,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: num] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ N ) ) ) ) ).
% odd_numeral
thf(fact_2998_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q3: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_2999_power3__eq__cube,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( times_times @ A @ ( times_times @ A @ A2 @ A2 ) @ A2 ) ) ) ).
% power3_eq_cube
thf(fact_3000_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_3001_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_3002_not__numeral__Bit0__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_3003_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q3: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_3004_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q3 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q3 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_3005_Suc__div__eq__add3__div,axiom,
! [M: nat,N: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N ) ) ).
% Suc_div_eq_add3_div
thf(fact_3006_Suc__mod__eq__add3__mod,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_3007_tan__45,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( one_one @ real ) ) ).
% tan_45
thf(fact_3008_mod__exhaust__less__4,axiom,
! [M: nat] :
( ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ nat ) )
| ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( one_one @ nat ) )
| ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
| ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ).
% mod_exhaust_less_4
thf(fact_3009_geometric__sums,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
=> ( sums @ A @ ( power_power @ A @ C2 ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ).
% geometric_sums
thf(fact_3010_power__half__series,axiom,
( sums @ real
@ ^ [N3: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N3 ) )
@ ( one_one @ real ) ) ).
% power_half_series
thf(fact_3011_lemma__tan__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ord_less @ real @ Y @ ( tan @ real @ X3 ) ) ) ) ).
% lemma_tan_total
thf(fact_3012_tan__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X ) ) ) ) ).
% tan_gt_zero
thf(fact_3013_lemma__tan__total1,axiom,
! [Y: real] :
? [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y ) ) ).
% lemma_tan_total1
thf(fact_3014_tan__mono__lt__eq,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_3015_tan__monotone_H,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ X )
= ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_3016_tan__monotone,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X ) ) ) ) ) ).
% tan_monotone
thf(fact_3017_tan__total,axiom,
! [Y: real] :
? [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y )
& ! [Y2: real] :
( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
& ( ord_less @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ Y2 )
= Y ) )
=> ( Y2 = X3 ) ) ) ).
% tan_total
thf(fact_3018_tan__minus__45,axiom,
( ( tan @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% tan_minus_45
thf(fact_3019_tan__inverse,axiom,
! [Y: real] :
( ( divide_divide @ real @ ( one_one @ real ) @ ( tan @ real @ Y ) )
= ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y ) ) ) ).
% tan_inverse
thf(fact_3020_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_3021_cos__60,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% cos_60
thf(fact_3022_sin__30,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% sin_30
thf(fact_3023_sums__if_H,axiom,
! [G: nat > real,X: real] :
( ( sums @ real @ G @ X )
=> ( sums @ real
@ ^ [N3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( zero_zero @ real ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ X ) ) ).
% sums_if'
thf(fact_3024_sums__if,axiom,
! [G: nat > real,X: real,F2: nat > real,Y: real] :
( ( sums @ real @ G @ X )
=> ( ( sums @ real @ F2 @ Y )
=> ( sums @ real
@ ^ [N3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( F2 @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ real @ X @ Y ) ) ) ) ).
% sums_if
thf(fact_3025_tan__pos__pi2__le,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_3026_tan__total__pos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y ) ) ) ).
% tan_total_pos
thf(fact_3027_tan__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( tan @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% tan_less_zero
thf(fact_3028_tan__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_3029_tan__mono__le,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) ) ) ) ) ).
% tan_mono_le
thf(fact_3030_tan__bound__pi2,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X ) ) @ ( one_one @ real ) ) ) ).
% tan_bound_pi2
thf(fact_3031_arctan__unique,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( tan @ real @ X )
= Y )
=> ( ( arctan @ Y )
= X ) ) ) ) ).
% arctan_unique
thf(fact_3032_arctan__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arctan @ ( tan @ real @ X ) )
= X ) ) ) ).
% arctan_tan
thf(fact_3033_arctan,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ ( arctan @ Y ) )
= Y ) ) ).
% arctan
thf(fact_3034_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_3035_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_3036_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_3037_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_3038_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_3039_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ? [Z: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z )
& ( ord_less @ real @ Z @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
& ( ( tan @ real @ Z )
= X ) ) ) ).
% tan_total_pi4
thf(fact_3040_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_3041_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_3042_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_3043_tan__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% tan_half
thf(fact_3044_time__vebt__buildup,axiom,
! [U2: nat] :
( ( U2
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ n ) )
=> ( ord_less_eq @ nat @ ( vEBT_V8346862874174094_d_u_p @ n ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ U2 ) ) ) ).
% time_vebt_buildup
thf(fact_3045_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
thf(fact_3046_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
thf(fact_3047_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
! [Va2: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
thf(fact_3048_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
! [Va2: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
thf(fact_3049_VEBT__internal_Obuildup__build__time,axiom,
! [N: nat] : ( ord_less @ nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) ).
% VEBT_internal.buildup_build_time
thf(fact_3050_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
( ( vEBT_V8346862874174094_d_u_p @ ( zero_zero @ nat ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
thf(fact_3051_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
( ( vEBT_V8646137997579335489_i_l_d @ ( zero_zero @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
thf(fact_3052_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
thf(fact_3053_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
thf(fact_3054_vebt__inst_Otime__vebt__buildup,axiom,
! [U2: nat,N: nat] :
( ( U2
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ U2 ) ) ) ).
% vebt_inst.time_vebt_buildup
thf(fact_3055_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_3056_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_3057_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_3058_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C2: nat > A,X: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( C2 @ N3 ) ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_3059_sin__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X ) ) ).
% sin_paired
thf(fact_3060_of__nat__fact,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( semiring_char_0_fact @ nat @ N ) )
= ( semiring_char_0_fact @ A @ N ) ) ) ).
% of_nat_fact
thf(fact_3061_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_3062_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_3063_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_3064_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral @ nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_3065_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_3066_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% pred_numeral_inc
thf(fact_3067_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_3068_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_3069_less__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_3070_less__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% less_numeral_Suc
thf(fact_3071_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_3072_le__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less_eq @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_3073_le__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% le_numeral_Suc
thf(fact_3074_diff__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( minus_minus @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_3075_diff__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( minus_minus @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_3076_minus__not__numeral__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ).
% minus_not_numeral_eq
thf(fact_3077_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_3078_fact__2,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% fact_2
thf(fact_3079_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_3080_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_3081_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_3082_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ).
% diff_numeral_special(6)
thf(fact_3083_signed__take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_3084_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_3085_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_3086_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_3087_diffs__of__real,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [F2: nat > real] :
( ( diffs @ A
@ ^ [N3: nat] : ( real_Vector_of_real @ A @ ( F2 @ N3 ) ) )
= ( ^ [N3: nat] : ( real_Vector_of_real @ A @ ( diffs @ real @ F2 @ N3 ) ) ) ) ) ).
% diffs_of_real
thf(fact_3088_fact__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: num] :
( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ K ) )
= ( times_times @ A @ ( numeral_numeral @ A @ K ) @ ( semiring_char_0_fact @ A @ ( pred_numeral @ K ) ) ) ) ) ).
% fact_numeral
thf(fact_3089_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one2 )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( P @ ( inc @ X3 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_3090_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus @ num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus @ num @ X @ Y ) ) ) ).
% add_inc
thf(fact_3091_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_3092_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_3093_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_3094_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_3095_diffs__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [C2: nat > A] :
( ( diffs @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( C2 @ N3 ) ) )
= ( ^ [N3: nat] : ( uminus_uminus @ A @ ( diffs @ A @ C2 @ N3 ) ) ) ) ) ).
% diffs_minus
thf(fact_3096_fact__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_mono
thf(fact_3097_fact__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M ) ) ) ) ).
% fact_dvd
thf(fact_3098_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_3099_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_3100_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_3101_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_3102_add__One,axiom,
! [X: num] :
( ( plus_plus @ num @ X @ one2 )
= ( inc @ X ) ) ).
% add_One
thf(fact_3103_fact__less__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ( ord_less @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ) ).
% fact_less_mono
thf(fact_3104_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_3105_fact__mod,axiom,
! [A: $tType] :
( ( ( linordered_semidom @ A )
& ( semidom_modulo @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M ) )
= ( zero_zero @ A ) ) ) ) ).
% fact_mod
thf(fact_3106_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_3107_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_3108_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_3109_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_3110_minus__numeral__inc__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% minus_numeral_inc_eq
thf(fact_3111_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% choose_dvd
thf(fact_3112_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C6: nat > A,N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) @ ( C6 @ ( suc @ N3 ) ) ) ) ) ) ).
% diffs_def
thf(fact_3113_termdiff__converges__all,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [X3: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ).
% termdiff_converges_all
thf(fact_3114_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_3115_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_3116_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X2: A] : ( minus_minus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_3117_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_3118_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M3: nat] :
( if @ A
@ ( M3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M3 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_3119_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_3120_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_3121_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_binomial
thf(fact_3122_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_3123_take__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_3124_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,K6: real,C2: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ K6 )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K6 )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) ) ) )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_3125_take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_3126_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_3127_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_3128_cos__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) @ ( power_power @ real @ X @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( cos @ real @ X ) ) ).
% cos_paired
thf(fact_3129_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N3 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_3130_cos__coeff__def,axiom,
( cos_coeff
= ( ^ [N3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( zero_zero @ real ) ) ) ) ).
% cos_coeff_def
thf(fact_3131_sin__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( sin @ real @ X )
= ( divide_divide @ real @ ( tan @ real @ X ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_3132_cos__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( cos @ real @ X )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_3133_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_3134_real__sqrt__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ X )
= ( sqrt @ Y ) )
= ( X = Y ) ) ).
% real_sqrt_eq_iff
thf(fact_3135_real__sqrt__zero,axiom,
( ( sqrt @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_sqrt_zero
thf(fact_3136_real__sqrt__eq__zero__cancel__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% real_sqrt_eq_zero_cancel_iff
thf(fact_3137_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% real_sqrt_less_iff
thf(fact_3138_real__sqrt__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% real_sqrt_le_iff
thf(fact_3139_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ).
% real_sqrt_eq_1_iff
thf(fact_3140_real__sqrt__one,axiom,
( ( sqrt @ ( one_one @ real ) )
= ( one_one @ real ) ) ).
% real_sqrt_one
thf(fact_3141_real__sqrt__gt__0__iff,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ).
% real_sqrt_gt_0_iff
thf(fact_3142_real__sqrt__lt__0__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% real_sqrt_lt_0_iff
thf(fact_3143_real__sqrt__ge__0__iff,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ).
% real_sqrt_ge_0_iff
thf(fact_3144_real__sqrt__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% real_sqrt_le_0_iff
thf(fact_3145_real__sqrt__gt__1__iff,axiom,
! [Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ).
% real_sqrt_gt_1_iff
thf(fact_3146_real__sqrt__lt__1__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ).
% real_sqrt_lt_1_iff
thf(fact_3147_real__sqrt__ge__1__iff,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ).
% real_sqrt_ge_1_iff
thf(fact_3148_real__sqrt__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ).
% real_sqrt_le_1_iff
thf(fact_3149_real__sqrt__abs2,axiom,
! [X: real] :
( ( sqrt @ ( times_times @ real @ X @ X ) )
= ( abs_abs @ real @ X ) ) ).
% real_sqrt_abs2
thf(fact_3150_real__sqrt__mult__self,axiom,
! [A2: real] :
( ( times_times @ real @ ( sqrt @ A2 ) @ ( sqrt @ A2 ) )
= ( abs_abs @ real @ A2 ) ) ).
% real_sqrt_mult_self
thf(fact_3151_sin__coeff__0,axiom,
( ( sin_coeff @ ( zero_zero @ nat ) )
= ( zero_zero @ real ) ) ).
% sin_coeff_0
thf(fact_3152_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_3153_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_3154_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_3155_dbl__inc__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bit1 @ K ) ) ) ) ).
% dbl_inc_simps(5)
thf(fact_3156_real__sqrt__four,axiom,
( ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% real_sqrt_four
thf(fact_3157_dbl__dec__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_3158_dbl__inc__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_3159_real__sqrt__abs,axiom,
! [X: real] :
( ( sqrt @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X ) ) ).
% real_sqrt_abs
thf(fact_3160_real__sqrt__pow2__iff,axiom,
! [X: real] :
( ( ( power_power @ real @ ( sqrt @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% real_sqrt_pow2_iff
thf(fact_3161_real__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( sqrt @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X ) ) ).
% real_sqrt_pow2
thf(fact_3162_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X: real,Y: real,Xa: 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 @ Xa @ ( 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 @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_squared_eq
thf(fact_3163_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_self
thf(fact_3164_fact__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ).
% fact_mono_nat
thf(fact_3165_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_less_mono
thf(fact_3166_real__sqrt__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_le_mono
thf(fact_3167_real__sqrt__minus,axiom,
! [X: real] :
( ( sqrt @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( sqrt @ X ) ) ) ).
% real_sqrt_minus
thf(fact_3168_real__sqrt__power,axiom,
! [X: real,K: nat] :
( ( sqrt @ ( power_power @ real @ X @ K ) )
= ( power_power @ real @ ( sqrt @ X ) @ K ) ) ).
% real_sqrt_power
thf(fact_3169_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_3170_real__sqrt__divide,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( divide_divide @ real @ X @ Y ) )
= ( divide_divide @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_divide
thf(fact_3171_fact__less__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_3172_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_gt_zero
thf(fact_3173_real__sqrt__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_zero
thf(fact_3174_real__sqrt__eq__zero__cancel,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sqrt @ X )
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ).
% real_sqrt_eq_zero_cancel
thf(fact_3175_real__sqrt__ge__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_one
thf(fact_3176_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_3177_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_3178_real__div__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( divide_divide @ real @ X @ ( sqrt @ X ) )
= ( sqrt @ X ) ) ) ).
% real_div_sqrt
thf(fact_3179_sqrt__add__le__add__sqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ X @ Y ) ) @ ( plus_plus @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_3180_dvd__fact,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ nat @ M @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_3181_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_3182_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_3183_sqrt2__less__2,axiom,
ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% sqrt2_less_2
thf(fact_3184_fact__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) )
= ( times_times @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_3185_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_3186_binomial__fact__lemma,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
= ( semiring_char_0_fact @ nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_3187_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_3188_real__less__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less @ real @ X @ ( sqrt @ Y ) ) ) ).
% real_less_rsqrt
thf(fact_3189_real__le__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less_eq @ real @ X @ ( sqrt @ Y ) ) ) ).
% real_le_rsqrt
thf(fact_3190_sqrt__le__D,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ Y )
=> ( ord_less_eq @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% sqrt_le_D
thf(fact_3191_binomial__altdef__nat,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_3192_real__sqrt__unique,axiom,
! [Y: real,X: real] :
( ( ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( sqrt @ X )
= Y ) ) ) ).
% real_sqrt_unique
thf(fact_3193_real__le__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_le_lsqrt
thf(fact_3194_lemma__real__divide__sqrt__less,axiom,
! [U2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ U2 )
=> ( ord_less @ real @ ( divide_divide @ real @ U2 @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ U2 ) ) ).
% lemma_real_divide_sqrt_less
thf(fact_3195_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= Y )
=> ( X
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel2
thf(fact_3196_real__sqrt__sum__squares__eq__cancel,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= X )
=> ( Y
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel
thf(fact_3197_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A2: real,C2: real,B2: real,D4: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( plus_plus @ real @ A2 @ C2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( plus_plus @ real @ B2 @ D4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ C2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_triangle_ineq
thf(fact_3198_real__sqrt__sum__squares__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq @ real @ Y @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_3199_real__sqrt__sum__squares__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_3200_sqrt__ge__absD,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( sqrt @ Y ) )
=> ( ord_less_eq @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) ) ).
% sqrt_ge_absD
thf(fact_3201_cos__45,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% cos_45
thf(fact_3202_sin__45,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% sin_45
thf(fact_3203_tan__60,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ).
% tan_60
thf(fact_3204_real__less__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_less_lsqrt
thf(fact_3205_sqrt__sum__squares__le__sum,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ X @ Y ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_3206_sqrt__even__pow2,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( sqrt @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sqrt_even_pow2
thf(fact_3207_real__sqrt__ge__abs1,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_3208_real__sqrt__ge__abs2,axiom,
! [Y: real,X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_3209_sqrt__sum__squares__le__sum__abs,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( abs_abs @ real @ X ) @ ( abs_abs @ real @ Y ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_3210_ln__sqrt,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( sqrt @ X ) )
= ( divide_divide @ real @ ( ln_ln @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% ln_sqrt
thf(fact_3211_arsinh__real__def,axiom,
( ( arsinh @ real )
= ( ^ [X2: real] : ( ln_ln @ real @ ( plus_plus @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_3212_cos__30,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% cos_30
thf(fact_3213_sin__60,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% sin_60
thf(fact_3214_complex__norm,axiom,
! [X: real,Y: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ X @ Y ) )
= ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_norm
thf(fact_3215_arsinh__real__aux,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% arsinh_real_aux
thf(fact_3216_real__sqrt__power__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( sqrt @ X ) @ N )
= ( power_power @ real @ X @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_3217_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X: real,Y: real,Xa: 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 @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_ge_zero
thf(fact_3218_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_3219_powr__half__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( sqrt @ X ) ) ) ).
% powr_half_sqrt
thf(fact_3220_tan__30,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ) ).
% tan_30
thf(fact_3221_cos__x__y__le__one,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( divide_divide @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( one_one @ real ) ) ).
% cos_x_y_le_one
thf(fact_3222_real__sqrt__sum__squares__less,axiom,
! [X: real,U2: real,Y: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ U2 @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( divide_divide @ real @ U2 @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U2 ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_3223_arcosh__real__def,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( arcosh @ real @ X )
= ( ln_ln @ real @ ( plus_plus @ real @ X @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arcosh_real_def
thf(fact_3224_cos__arctan,axiom,
! [X: real] :
( ( cos @ real @ ( arctan @ X ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_arctan
thf(fact_3225_sin__arctan,axiom,
! [X: real] :
( ( sin @ real @ ( arctan @ X ) )
= ( divide_divide @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arctan
thf(fact_3226_sqrt__sum__squares__half__less,axiom,
! [X: real,U2: real,Y: real] :
( ( ord_less @ real @ X @ ( divide_divide @ real @ U2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ U2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U2 ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_3227_sin__cos__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) )
=> ( ( sin @ real @ X )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( cos @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_cos_sqrt
thf(fact_3228_arctan__half,axiom,
( arctan
= ( ^ [X2: real] : ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ X2 @ ( plus_plus @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_3229_binomial__code,axiom,
( binomial
= ( ^ [N3: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N3 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N3 @ ( minus_minus @ nat @ N3 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N3 @ K3 ) @ ( one_one @ nat ) ) @ N3 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_3230_cos__arcsin,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_arcsin
thf(fact_3231_sin__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ Y ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_3232_sin__arccos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_arccos
thf(fact_3233_time__vebt__member,axiom,
! [T2: vEBT_VEBT,U2: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( U2
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ n ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U2 ) ) ) ) ) ) ) ).
% time_vebt_member
thf(fact_3234_concat__bit__Suc,axiom,
! [N: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
= ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_3235_arcsin__0,axiom,
( ( arcsin @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% arcsin_0
thf(fact_3236_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_3237_arccos__1,axiom,
( ( arccos @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% arccos_1
thf(fact_3238_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N @ K @ L ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ).
% concat_bit_nonnegative_iff
thf(fact_3239_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less @ int @ ( bit_concat_bit @ N @ K @ L ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ).
% concat_bit_negative_iff
thf(fact_3240_concat__bit__of__zero__2,axiom,
! [N: nat,K: int] :
( ( bit_concat_bit @ N @ K @ ( zero_zero @ int ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% concat_bit_of_zero_2
thf(fact_3241_cos__arccos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ) ).
% cos_arccos
thf(fact_3242_sin__arcsin,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ).
% sin_arcsin
thf(fact_3243_arccos__0,axiom,
( ( arccos @ ( zero_zero @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arccos_0
thf(fact_3244_arcsin__1,axiom,
( ( arcsin @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arcsin_1
thf(fact_3245_arcsin__minus__1,axiom,
( ( arcsin @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% arcsin_minus_1
thf(fact_3246_VEBT__internal_Ovalid__tree__deg__neq__0,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% VEBT_internal.valid_tree_deg_neq_0
thf(fact_3247_VEBT__internal_Ovalid__0__not,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% VEBT_internal.valid_0_not
thf(fact_3248_concat__bit__assoc,axiom,
! [N: nat,K: int,M: nat,L: int,R3: int] :
( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R3 ) )
= ( bit_concat_bit @ ( plus_plus @ nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R3 ) ) ).
% concat_bit_assoc
thf(fact_3249_concat__bit__eq__iff,axiom,
! [N: nat,K: int,L: int,R3: int,S: int] :
( ( ( bit_concat_bit @ N @ K @ L )
= ( bit_concat_bit @ N @ R3 @ S ) )
= ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2584673776208193580ke_bit @ int @ N @ R3 ) )
& ( L = S ) ) ) ).
% concat_bit_eq_iff
thf(fact_3250_concat__bit__take__bit__eq,axiom,
! [N: nat,B2: int] :
( ( bit_concat_bit @ N @ ( bit_se2584673776208193580ke_bit @ int @ N @ B2 ) )
= ( bit_concat_bit @ N @ B2 ) ) ).
% concat_bit_take_bit_eq
thf(fact_3251_VEBT__internal_Odeg__not__0,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% VEBT_internal.deg_not_0
thf(fact_3252_arccos__le__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_3253_arccos__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ).
% arccos_le_mono
thf(fact_3254_arccos__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
& ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) ) )
=> ( ( ( arccos @ X )
= ( arccos @ Y ) )
= ( X = Y ) ) ) ).
% arccos_eq_iff
thf(fact_3255_arcsin__le__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_3256_arcsin__minus,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( arcsin @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( arcsin @ X ) ) ) ) ) ).
% arcsin_minus
thf(fact_3257_arcsin__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% arcsin_le_mono
thf(fact_3258_arcsin__eq__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ( arcsin @ X )
= ( arcsin @ Y ) )
= ( X = Y ) ) ) ) ).
% arcsin_eq_iff
thf(fact_3259_arccos__lbound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) ) ) ) ).
% arccos_lbound
thf(fact_3260_arccos__less__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_3261_arccos__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ).
% arccos_less_mono
thf(fact_3262_arccos__ubound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_3263_arccos__cos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( arccos @ ( cos @ real @ X ) )
= X ) ) ) ).
% arccos_cos
thf(fact_3264_arcsin__less__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_3265_arcsin__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% arcsin_less_mono
thf(fact_3266_cos__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ).
% cos_arccos_abs
thf(fact_3267_arccos__cos__eq__abs,axiom,
! [Theta: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Theta ) @ pi )
=> ( ( arccos @ ( cos @ real @ Theta ) )
= ( abs_abs @ real @ Theta ) ) ) ).
% arccos_cos_eq_abs
thf(fact_3268_arccos__lt__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less @ real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_3269_arccos__bounded,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less_eq @ real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_3270_sin__arccos__nonzero,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X ) )
!= ( zero_zero @ real ) ) ) ) ).
% sin_arccos_nonzero
thf(fact_3271_arccos__cos2,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( arccos @ ( cos @ real @ X ) )
= ( uminus_uminus @ real @ X ) ) ) ) ).
% arccos_cos2
thf(fact_3272_arccos__minus,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X ) )
= ( minus_minus @ real @ pi @ ( arccos @ X ) ) ) ) ) ).
% arccos_minus
thf(fact_3273_cos__arcsin__nonzero,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X ) )
!= ( zero_zero @ real ) ) ) ) ).
% cos_arcsin_nonzero
thf(fact_3274_arccos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less_eq @ real @ ( arccos @ Y ) @ pi )
& ( ( cos @ real @ ( arccos @ Y ) )
= Y ) ) ) ) ).
% arccos
thf(fact_3275_arccos__minus__abs,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X ) )
= ( minus_minus @ real @ pi @ ( arccos @ X ) ) ) ) ).
% arccos_minus_abs
thf(fact_3276_arccos__le__pi2,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_3277_arcsin__lt__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_3278_arcsin__bounded,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_3279_arcsin__ubound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_3280_arcsin__lbound,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% arcsin_lbound
thf(fact_3281_arcsin__sin,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arcsin @ ( sin @ real @ X ) )
= X ) ) ) ).
% arcsin_sin
thf(fact_3282_arcsin,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin
thf(fact_3283_arcsin__pi,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq @ real @ ( arcsin @ Y ) @ pi )
& ( ( sin @ real @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
% arcsin_pi
thf(fact_3284_arcsin__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X ) @ Y )
= ( ord_less_eq @ real @ X @ ( sin @ real @ Y ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_3285_le__arcsin__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ Y @ ( arcsin @ X ) )
= ( ord_less_eq @ real @ ( sin @ real @ Y ) @ X ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_3286_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K2: int] :
( ( arccos @ ( cos @ real @ Theta ) )
!= ( abs_abs @ real @ ( minus_minus @ real @ Theta @ ( times_times @ real @ ( ring_1_of_int @ real @ K2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) ) ) ) ).
% arccos_cos_eq_abs_2pi
thf(fact_3287_vebt__space__linear__bound,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ n ) ) ) ) ).
% vebt_space_linear_bound
thf(fact_3288_time__vebt__insert,axiom,
! [T2: vEBT_VEBT,U2: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( U2
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ n ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U2 ) ) ) ) ) ) ) ).
% time_vebt_insert
thf(fact_3289_time__vebt__succ,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_s_u_c_c @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ n ) ) ) ) ) ) ) ).
% time_vebt_succ
thf(fact_3290_time__vebt__pred,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_p_r_e_d @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ n ) ) ) ) ) ) ) ).
% time_vebt_pred
thf(fact_3291_time__vebt__delete,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ n ) ) ) ) ) ) ) ).
% time_vebt_delete
thf(fact_3292_vebt__inst_Ovebt__space__linear__bound,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% vebt_inst.vebt_space_linear_bound
thf(fact_3293_VEBT__internal_Ospace__bound,axiom,
! [T2: vEBT_VEBT,N: nat,U2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U2
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ U2 ) ) ) ) ).
% VEBT_internal.space_bound
thf(fact_3294_vebt__inst_Otime__vebt__delete,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% vebt_inst.time_vebt_delete
thf(fact_3295_vebt__inst_Otime__vebt__pred,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_p_r_e_d @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% vebt_inst.time_vebt_pred
thf(fact_3296_vebt__inst_Otime__vebt__succ,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_s_u_c_c @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% vebt_inst.time_vebt_succ
thf(fact_3297_member__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U2: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U2
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U2 ) ) ) ) ) ) ) ).
% member_bound_size_univ
thf(fact_3298_succ__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U2: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U2
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_s_u_c_c @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U2 ) ) ) ) ) ) ) ).
% succ_bound_size_univ
thf(fact_3299_insert__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U2: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U2
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U2 ) ) ) ) ) ) ) ).
% insert_bound_size_univ
thf(fact_3300_pred__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U2: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U2
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_p_r_e_d @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U2 ) ) ) ) ) ) ) ).
% pred_bound_size_univ
thf(fact_3301_VEBT__internal_Odelete__bound__size__univ,axiom,
! [T2: vEBT_VEBT,N: nat,U2: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U2
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U2 ) ) ) ) ) ) ) ).
% VEBT_internal.delete_bound_size_univ
thf(fact_3302_invar__vebt__insert,axiom,
! [T2: vEBT_VEBT,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 ) ) ) ).
% invar_vebt_insert
thf(fact_3303_VEBT__internal_Ospace__2__pow__bound,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_space2 @ T2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ real ) ) ) ) ) ).
% VEBT_internal.space_2_pow_bound
thf(fact_3304_VEBT__internal_Ocnt__bound,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ T2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.cnt_bound
thf(fact_3305_succ__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U2: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U2
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_s_u_c_c2 @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U2 ) ) ) ) ) ) ).
% succ_bound_size_univ'
thf(fact_3306_pred__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U2: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U2
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_T_p_r_e_d2 @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U2 ) ) ) ) ) ) ).
% pred_bound_size_univ'
thf(fact_3307_VEBT__internal_Ocnt__non__neg,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( vEBT_VEBT_cnt @ T2 ) ) ).
% VEBT_internal.cnt_non_neg
thf(fact_3308_VEBT__internal_Ospace__space_H,axiom,
! [T2: vEBT_VEBT] : ( ord_less @ nat @ ( vEBT_VEBT_space @ T2 ) @ ( vEBT_VEBT_space2 @ T2 ) ) ).
% VEBT_internal.space_space'
thf(fact_3309_VEBT__internal_Ospace__cnt,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_space2 @ T2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_VEBT_cnt @ T2 ) ) ) ).
% VEBT_internal.space_cnt
thf(fact_3310_VEBT__internal_Ospace_H__bound,axiom,
! [T2: vEBT_VEBT,N: nat,U2: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U2
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_space2 @ T2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ U2 ) ) ) ) ).
% VEBT_internal.space'_bound
thf(fact_3311_VEBT__internal_Ocnt__bound_H,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ T2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( minus_minus @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ real ) ) ) ) ) ).
% VEBT_internal.cnt_bound'
thf(fact_3312_VEBT__internal_Ovalid__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 ) ) ) ).
% VEBT_internal.valid_pres_insert
thf(fact_3313_VEBT__internal_Odelete__bound__size__univ_H,axiom,
! [T2: vEBT_VEBT,N: nat,U2: real,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( U2
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X ) ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U2 ) ) ) ) ) ) ).
% VEBT_internal.delete_bound_size_univ'
thf(fact_3314_VEBT__internal_Ot__build__cnt,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) @ ( times_times @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ).
% VEBT_internal.t_build_cnt
thf(fact_3315_VEBT__internal_Ot__buildup__cnt,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_V8346862874174094_d_u_p @ N ) ) @ ( times_times @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ).
% VEBT_internal.t_buildup_cnt
thf(fact_3316_succ__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% succ_bound_height
thf(fact_3317_invar__vebt__buildup,axiom,
( ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ n ) @ n )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ n ) ) ).
% invar_vebt_buildup
thf(fact_3318_VEBT__internal_Odelete__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% VEBT_internal.delete_bound_height'
thf(fact_3319_VEBT__internal_Obuildup__gives__valid,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% VEBT_internal.buildup_gives_valid
thf(fact_3320_vebt__inst_Oinvar__vebt__buildup,axiom,
! [N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% vebt_inst.invar_vebt_buildup
thf(fact_3321_pred__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d2 @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% pred_bound_height'
thf(fact_3322_succ_H__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_s_u_c_c2 @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% succ'_bound_height
thf(fact_3323_VEBT__internal_Otwo__powr__height__bound__deg,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% VEBT_internal.two_powr_height_bound_deg
thf(fact_3324_VEBT__internal_Ocount__buildup,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% VEBT_internal.count_buildup
thf(fact_3325_VEBT__internal_Oheigt__uplog__rel,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ T2 ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% VEBT_internal.heigt_uplog_rel
thf(fact_3326_VEBT__internal_Ocount__buildup_H,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( semiring_1_of_nat @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% VEBT_internal.count_buildup'
thf(fact_3327_member__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% member_bound_height
thf(fact_3328_VEBT__internal_Oheight__double__log__univ__size,axiom,
! [U2: real,Deg: nat,T2: vEBT_VEBT] :
( ( U2
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_invar_vebt @ T2 @ Deg )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( vEBT_VEBT_height @ T2 ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ U2 ) ) ) ) ) ) ).
% VEBT_internal.height_double_log_univ_size
thf(fact_3329_VEBT__internal_Odelete__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.delete_bound_height
thf(fact_3330_pred__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_p_r_e_d @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% pred_bound_height
thf(fact_3331_insert__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X ) @ ( times_times @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% insert_bound_height
thf(fact_3332_cot__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cot @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% cot_less_zero
thf(fact_3333_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( if @ A
@ ( K3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ L2 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_3334_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_3335_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq1
thf(fact_3336_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_3337_nat__of__bool,axiom,
! [P: $o] :
( ( nat2 @ ( zero_neq_one_of_bool @ int @ P ) )
= ( zero_neq_one_of_bool @ nat @ P ) ) ).
% nat_of_bool
thf(fact_3338_of__bool__eq__0__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( zero_zero @ A ) )
= ~ P ) ) ).
% of_bool_eq_0_iff
thf(fact_3339_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_3340_of__bool__less__eq__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P: $o,Q: $o] :
( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( P
=> Q ) ) ) ).
% of_bool_less_eq_iff
thf(fact_3341_of__bool__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P: $o,Q: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( ~ P
& Q ) ) ) ).
% of_bool_less_iff
thf(fact_3342_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( one_one @ A ) )
= P ) ) ).
% of_bool_eq_1_iff
thf(fact_3343_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_3344_of__nat__of__bool,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o] :
( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% of_nat_of_bool
thf(fact_3345_abs__bool__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: $o] :
( ( abs_abs @ A @ ( zero_neq_one_of_bool @ A @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% abs_bool_eq
thf(fact_3346_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) ) ) ).
% exp_less_mono
thf(fact_3347_exp__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% exp_less_cancel_iff
thf(fact_3348_of__int__of__bool,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: $o] :
( ( ring_1_of_int @ A @ ( zero_neq_one_of_bool @ int @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% of_int_of_bool
thf(fact_3349_exp__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% exp_le_cancel_iff
thf(fact_3350_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_3351_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_3352_zero__less__of__bool__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) )
= P ) ) ).
% zero_less_of_bool_iff
thf(fact_3353_of__bool__less__one__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) )
= ~ P ) ) ).
% of_bool_less_one_iff
thf(fact_3354_of__bool__not__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: $o] :
( ( zero_neq_one_of_bool @ A @ ~ P )
= ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ).
% of_bool_not_iff
thf(fact_3355_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_3356_gbinomial__0_I2_J,axiom,
! [B: $tType] :
( ( ( semiring_char_0 @ B )
& ( semidom_divide @ B ) )
=> ! [K: nat] :
( ( gbinomial @ B @ ( zero_zero @ B ) @ ( suc @ K ) )
= ( zero_zero @ B ) ) ) ).
% gbinomial_0(2)
thf(fact_3357_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_3358_gbinomial__Suc0,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% gbinomial_Suc0
thf(fact_3359_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp @ real @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% exp_eq_one_iff
thf(fact_3360_cot__pi,axiom,
( ( cot @ real @ pi )
= ( zero_zero @ real ) ) ).
% cot_pi
thf(fact_3361_exp__less__one__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% exp_less_one_iff
thf(fact_3362_one__less__exp__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% one_less_exp_iff
thf(fact_3363_exp__le__one__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% exp_le_one_iff
thf(fact_3364_one__le__exp__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( exp @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% one_le_exp_iff
thf(fact_3365_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_3366_exp__ln,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( exp @ real @ ( ln_ln @ real @ X ) )
= X ) ) ).
% exp_ln
thf(fact_3367_exp__ln__iff,axiom,
! [X: real] :
( ( ( exp @ real @ ( ln_ln @ real @ X ) )
= X )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% exp_ln_iff
thf(fact_3368_odd__of__bool__self,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [P5: $o] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_neq_one_of_bool @ A @ P5 ) ) )
= P5 ) ) ).
% odd_of_bool_self
thf(fact_3369_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_3370_cot__npi,axiom,
! [N: nat] :
( ( cot @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% cot_npi
thf(fact_3371_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_3372_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_3373_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_3374_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_3375_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_3376_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_of_exp
thf(fact_3377_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_3378_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_3379_norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ X ) ) @ ( exp @ real @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ) ).
% norm_exp
thf(fact_3380_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
=> ( ord_less @ real @ X @ Y ) ) ).
% exp_less_cancel
thf(fact_3381_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P5: $o,Q3: $o] :
( ( ( zero_neq_one_of_bool @ A @ P5 )
= ( zero_neq_one_of_bool @ A @ Q3 ) )
= ( P5 = Q3 ) ) ) ).
% of_bool_eq_iff
thf(fact_3382_of__nat__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat,K: nat] :
( ( semiring_1_of_nat @ A @ ( gbinomial @ nat @ N @ K ) )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% of_nat_gbinomial
thf(fact_3383_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_3384_of__bool__conj,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
& Q ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_conj
thf(fact_3385_subset__decode__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% subset_decode_imp_le
thf(fact_3386_exp__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X3: real] :
( ( exp @ real @ X3 )
= Y ) ) ).
% exp_total
thf(fact_3387_exp__gt__zero,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_gt_zero
thf(fact_3388_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_less_zero
thf(fact_3389_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_ge_zero
thf(fact_3390_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_le_zero
thf(fact_3391_zero__less__eq__of__bool,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% zero_less_eq_of_bool
thf(fact_3392_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P5: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
= ( ~ ( ( P5
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P5
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_3393_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P5: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
= ( ( P5
=> ( P @ ( one_one @ A ) ) )
& ( ~ P5
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_3394_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P6: $o] : ( if @ A @ P6 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_3395_of__bool__less__eq__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) ) ).
% of_bool_less_eq_one
thf(fact_3396_binomial__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat,K: nat] :
( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% binomial_gbinomial
thf(fact_3397_exp__gt__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X ) ) ) ).
% exp_gt_one
thf(fact_3398_exp__ge__add__one__self,axiom,
! [X: real] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( exp @ real @ X ) ) ).
% exp_ge_add_one_self
thf(fact_3399_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_3400_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_3401_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_3402_exp__ge__add__one__self__aux,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( exp @ real @ X ) ) ) ).
% exp_ge_add_one_self_aux
thf(fact_3403_lemma__exp__total,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ Y )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( minus_minus @ real @ Y @ ( one_one @ real ) ) )
& ( ( exp @ real @ X3 )
= Y ) ) ) ).
% lemma_exp_total
thf(fact_3404_ln__ge__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( exp @ real @ Y ) @ X ) ) ) ).
% ln_ge_iff
thf(fact_3405_ln__x__over__x__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( ln_ln @ real @ Y ) @ Y ) @ ( divide_divide @ real @ ( ln_ln @ real @ X ) @ X ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_3406_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ A2 @ ( gbinomial @ A @ A2 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_3407_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_3408_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A2 @ K ) )
= ( times_times @ A @ A2 @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_3409_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,A2: A] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_3410_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X2: A,A5: A] :
( if @ A
@ ( X2
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A5 @ ( ln_ln @ A @ X2 ) ) ) ) ) ) ) ).
% powr_def
thf(fact_3411_exp__le,axiom,
ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ).
% exp_le
thf(fact_3412_of__bool__odd__eq__mod__2,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_bool_odd_eq_mod_2
thf(fact_3413_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_3414_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) )
= ( times_times @ A @ A2 @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorption
thf(fact_3415_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M: nat,A2: A] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( times_times @ A @ ( gbinomial @ A @ A2 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_3416_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_3417_exp__half__le2,axiom,
ord_less_eq @ real @ ( exp @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% exp_half_le2
thf(fact_3418_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A2: A] :
( ! [A6: A] :
( ( ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ A6 ) )
=> ( ! [A6: A,B6: $o] :
( ( P @ A6 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B6 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B6 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% bits_induct
thf(fact_3419_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_3420_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_3421_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ K ) )
= ( 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 @ K ) ) @ ( one_one @ A ) ) @ N ) ) ) ) ).
% gbinomial_index_swap
thf(fact_3422_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: 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 ) @ A5 ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_3423_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_3424_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% exp_mod_exp
thf(fact_3425_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_minus
thf(fact_3426_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A2 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_3427_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: 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 @ A5 ) @ K3 ) ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_3428_exp__bound__half,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 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% exp_bound_half
thf(fact_3429_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_3430_exp__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% exp_bound
thf(fact_3431_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_3432_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M3: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_3433_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% one_xor_eq
thf(fact_3434_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% xor_one_eq
thf(fact_3435_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_3436_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_3437_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_3438_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_3439_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_3440_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) )
& ( ord_less_eq @ nat @ N @ M ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_3441_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_3442_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A2 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_3443_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_3444_cot__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X ) ) ) ) ).
% cot_gt_zero
thf(fact_3445_exp__lower__Taylor__quadratic,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( divide_divide @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ X ) ) ) ).
% exp_lower_Taylor_quadratic
thf(fact_3446_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq2
thf(fact_3447_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_3448_tan__cot_H,axiom,
! [X: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) )
= ( cot @ real @ X ) ) ).
% tan_cot'
thf(fact_3449_tanh__real__altdef,axiom,
( ( tanh @ real )
= ( ^ [X2: real] : ( divide_divide @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_3450_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X2: nat] :
( collect @ nat
@ ^ [N3: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_3451_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R3: A,A2: A,B2: A,C2: A,D4: A] :
( ( R3
!= ( zero_zero @ A ) )
=> ( ( ( A2 = B2 )
& ( C2 != D4 ) )
=> ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R3 @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R3 @ D4 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_3452_arctan__def,axiom,
( arctan
= ( ^ [Y4: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
& ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X2 )
= Y4 ) ) ) ) ) ).
% arctan_def
thf(fact_3453_arcsin__def,axiom,
( arcsin
= ( ^ [Y4: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X2 )
= Y4 ) ) ) ) ) ).
% arcsin_def
thf(fact_3454_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_3455_num_Osize__gen_I3_J,axiom,
! [X32: num] :
( ( size_num @ ( bit1 @ X32 ) )
= ( plus_plus @ nat @ ( size_num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(3)
thf(fact_3456_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_3457_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_3458_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_3459_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_3460_take__bit__minus__one__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_3461_nat__mask__eq,axiom,
! [N: nat] :
( ( nat2 @ ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% nat_mask_eq
thf(fact_3462_of__nat__mask__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% of_nat_mask_eq
thf(fact_3463_of__int__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( ring_1_of_int @ A @ ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% of_int_mask_eq
thf(fact_3464_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% less_eq_mask
thf(fact_3465_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% mask_nonnegative_int
thf(fact_3466_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_3467_ln__real__def,axiom,
( ( ln_ln @ real )
= ( ^ [X2: real] :
( the @ real
@ ^ [U: real] :
( ( exp @ real @ U )
= X2 ) ) ) ) ).
% ln_real_def
thf(fact_3468_suminf__def,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( suminf @ A )
= ( ^ [F3: nat > A] : ( the @ A @ ( sums @ A @ F3 ) ) ) ) ) ).
% suminf_def
thf(fact_3469_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_3470_ln__neg__is__const,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ X )
= ( the @ real
@ ^ [X2: real] : $false ) ) ) ).
% ln_neg_is_const
thf(fact_3471_take__bit__not__eq__mask__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) )
= ( minus_minus @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ).
% take_bit_not_eq_mask_diff
thf(fact_3472_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ) ).
% take_bit_eq_mask_iff
thf(fact_3473_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_3474_take__bit__not__mask__eq__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_not_mask_eq_0
thf(fact_3475_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_3476_mask__nat__less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% mask_nat_less_exp
thf(fact_3477_arccos__def,axiom,
( arccos
= ( ^ [Y4: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ pi )
& ( ( cos @ real @ X2 )
= Y4 ) ) ) ) ) ).
% arccos_def
thf(fact_3478_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_3479_add__0__iff,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [B2: A,A2: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_3480_mask__nat__def,axiom,
( ( bit_se2239418461657761734s_mask @ nat )
= ( ^ [N3: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) ) ).
% mask_nat_def
thf(fact_3481_mask__half__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_se2239418461657761734s_mask @ int @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% mask_half_int
thf(fact_3482_mask__int__def,axiom,
( ( bit_se2239418461657761734s_mask @ int )
= ( ^ [N3: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ int ) ) ) ) ).
% mask_int_def
thf(fact_3483_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N3: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_3484_minus__exp__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% minus_exp_eq_not_mask
thf(fact_3485_pi__half,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
= ( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X2 )
= ( zero_zero @ real ) ) ) ) ) ).
% pi_half
thf(fact_3486_pi__def,axiom,
( pi
= ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) )
@ ( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X2 )
= ( zero_zero @ real ) ) ) ) ) ) ).
% pi_def
thf(fact_3487_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_3488_num_Osize__gen_I2_J,axiom,
! [X22: num] :
( ( size_num @ ( bit0 @ X22 ) )
= ( plus_plus @ nat @ ( size_num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(2)
thf(fact_3489_the__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( the @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_3490_the__eq__trivial,axiom,
! [A: $tType,A2: A] :
( ( the @ A
@ ^ [X2: A] : X2 = A2 )
= A2 ) ).
% the_eq_trivial
thf(fact_3491_the__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ( the @ A @ P )
= A2 ) ) ) ).
% the_equality
thf(fact_3492_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_3493_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_3494_sgn__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ).
% sgn_sgn
thf(fact_3495_sgn__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_3496_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_3497_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_3498_idom__abs__sgn__class_Osgn__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ A2 ) ) ) ) ).
% idom_abs_sgn_class.sgn_minus
thf(fact_3499_power__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( sgn_sgn @ A @ ( power_power @ A @ A2 @ N ) )
= ( power_power @ A @ ( sgn_sgn @ A @ A2 ) @ N ) ) ) ).
% power_sgn
thf(fact_3500_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_greater
thf(fact_3501_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_3502_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_3503_bit__numeral__Bit0__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N ) ) ) ).
% bit_numeral_Bit0_Suc_iff
thf(fact_3504_bit__numeral__Bit1__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N ) ) ) ).
% bit_numeral_Bit1_Suc_iff
thf(fact_3505_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_3506_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) )
= ( zero_neq_one_of_bool @ A
@ ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_3507_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( abs_abs @ A @ A2 ) )
= ( zero_neq_one_of_bool @ A
@ ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_3508_sgn__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( zero_neq_one_of_bool @ A
@ ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_abs
thf(fact_3509_sgn__mult__dvd__iff,axiom,
! [R3: int,L: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R3 ) @ L ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R3
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_3510_mult__sgn__dvd__iff,axiom,
! [L: int,R3: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R3 ) ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R3
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_3511_dvd__sgn__mult__iff,axiom,
! [L: int,R3: int,K: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R3 ) @ K ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R3
= ( zero_zero @ int ) ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_3512_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R3: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R3 ) ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R3
= ( zero_zero @ int ) ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_3513_signed__take__bit__nonnegative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% signed_take_bit_nonnegative_iff
thf(fact_3514_signed__take__bit__negative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ).
% signed_take_bit_negative_iff
thf(fact_3515_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_3516_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_3517_bit__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W2: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W2 ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W2 ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_numeral_simps(2)
thf(fact_3518_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_3519_bit__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W2: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W2 ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W2 ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_numeral_simps(3)
thf(fact_3520_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_3521_bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( zero_zero @ nat ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% bit_0
thf(fact_3522_bit__minus__numeral__int_I1_J,axiom,
! [W2: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ ( pred_numeral @ N ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_3523_bit__minus__numeral__int_I2_J,axiom,
! [W2: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_3524_bit__mod__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( N
= ( zero_zero @ nat ) )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% bit_mod_2_iff
thf(fact_3525_bit__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
& ( M != N ) ) ) ) ).
% bit_unset_bit_iff
thf(fact_3526_bit__disjunctive__add__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A,N: nat] :
( ! [N2: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_3527_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_mult
thf(fact_3528_bit__of__nat__iff__bit,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N )
= ( bit_se5641148757651400278ts_bit @ nat @ M @ N ) ) ) ).
% bit_of_nat_iff_bit
thf(fact_3529_same__sgn__sgn__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A2 ) )
=> ( ( sgn_sgn @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ) ).
% same_sgn_sgn_add
thf(fact_3530_bit__numeral__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N )
= ( bit_se5641148757651400278ts_bit @ nat @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% bit_numeral_iff
thf(fact_3531_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_3532_sgn__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_eq_0_iff
thf(fact_3533_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_3534_bit__xor__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
!= ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ).
% bit_xor_iff
thf(fact_3535_bit__not__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_not_int_iff
thf(fact_3536_bit__xor__int__iff,axiom,
! [K: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
!= ( bit_se5641148757651400278ts_bit @ int @ L @ N ) ) ) ).
% bit_xor_int_iff
thf(fact_3537_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_3538_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: num] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% bit_numeral_simps(1)
thf(fact_3539_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_3540_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) @ N )
= ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% bit_take_bit_iff
thf(fact_3541_sgn__not__eq__imp,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
!= ( sgn_sgn @ A @ A2 ) )
=> ( ( ( sgn_sgn @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( ( ( sgn_sgn @ A @ B2 )
!= ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ B2 ) ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_3542_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_3543_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_3544_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_3545_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) )
= A2 ) ) ).
% abs_mult_sgn
thf(fact_3546_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( abs_abs @ A @ A2 ) )
= A2 ) ) ).
% sgn_mult_abs
thf(fact_3547_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_3548_same__sgn__abs__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A2 ) )
=> ( ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% same_sgn_abs_add
thf(fact_3549_int__sgnE,axiom,
! [K: int] :
~ ! [N2: nat,L3: int] :
( K
!= ( times_times @ int @ ( sgn_sgn @ int @ L3 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_sgnE
thf(fact_3550_signed__take__bit__eq__if__positive,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
=> ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ).
% signed_take_bit_eq_if_positive
thf(fact_3551_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_1_pos
thf(fact_3552_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( A2
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( zero_zero @ A ) ) )
& ( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_3553_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( minus_minus @ int @ ( uminus_uminus @ int @ K ) @ ( one_one @ int ) ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_not_int_iff'
thf(fact_3554_sgn__mod,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ~ ( dvd_dvd @ int @ L @ K )
=> ( ( sgn_sgn @ int @ ( modulo_modulo @ int @ K @ L ) )
= ( sgn_sgn @ int @ L ) ) ) ) ).
% sgn_mod
thf(fact_3555_flip__bit__eq__if,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N3: nat,A5: A] : ( if @ ( nat > A > A ) @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ N3 ) @ ( bit_se2638667681897837118et_bit @ A ) @ ( bit_se5668285175392031749et_bit @ A ) @ N3 @ A5 ) ) ) ) ).
% flip_bit_eq_if
thf(fact_3556_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_3557_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X2: A] :
( if @ A
@ ( X2
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_3558_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I4: int] :
( if @ int
@ ( I4
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I4 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_3559_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_3560_div__sgn__abs__cancel,axiom,
! [V2: int,K: int,L: int] :
( ( V2
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ V2 ) @ ( abs_abs @ int @ K ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ V2 ) @ ( abs_abs @ int @ L ) ) )
= ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_3561_bit__imp__take__bit__positive,axiom,
! [N: nat,M: nat,K: int] :
( ( ord_less @ nat @ N @ M )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_3562_div__dvd__sgn__abs,axiom,
! [L: int,K: int] :
( ( dvd_dvd @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( sgn_sgn @ int @ L ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_dvd_sgn_abs
thf(fact_3563_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L ) @ N )
= ( ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_3564_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ K ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) ) @ N ) ) ).
% bit_minus_int_iff
thf(fact_3565_signed__take__bit__eq__concat__bit,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( bit_concat_bit @ N3 @ K3 @ ( uminus_uminus @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) ) ) ) ) ).
% signed_take_bit_eq_concat_bit
thf(fact_3566_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,A2: A] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_3567_bit__Suc,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% bit_Suc
thf(fact_3568_stable__imp__bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% stable_imp_bit_iff_odd
thf(fact_3569_bit__iff__idd__imp__stable,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
=> ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 ) ) ) ).
% bit_iff_idd_imp_stable
thf(fact_3570_int__bit__bound,axiom,
! [K: int] :
~ ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M2 )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) ) ) ).
% int_bit_bound
thf(fact_3571_theI,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_3572_theI_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( P @ ( the @ A @ P ) ) ) ).
% theI'
thf(fact_3573_theI2,axiom,
! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ) ).
% theI2
thf(fact_3574_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P3: $o,X2: A,Y4: A] :
( the @ A
@ ^ [Z7: A] :
( ( P3
=> ( Z7 = X2 ) )
& ( ~ P3
=> ( Z7 = Y4 ) ) ) ) ) ) ).
% If_def
thf(fact_3575_the1I2,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ).
% the1I2
thf(fact_3576_the1__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( ( P @ A2 )
=> ( ( the @ A @ P )
= A2 ) ) ) ).
% the1_equality
thf(fact_3577_bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N3: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ) ).
% bit_iff_odd
thf(fact_3578_bit__not__iff__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ N )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% bit_not_iff_eq
thf(fact_3579_bit__int__def,axiom,
( ( bit_se5641148757651400278ts_bit @ int )
= ( ^ [K3: int,N3: nat] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% bit_int_def
thf(fact_3580_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_3581_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_3582_div__noneq__sgn__abs,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ K @ L )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L @ K ) ) ) ) ) ) ).
% div_noneq_sgn_abs
thf(fact_3583_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ! [J2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ J2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_3584_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N3: nat] :
( ( ( N3
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) )
& ( ( N3
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_3585_floor__real__def,axiom,
( ( archim6421214686448440834_floor @ real )
= ( ^ [X2: real] :
( the @ int
@ ^ [Z7: int] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z7 ) @ X2 )
& ( ord_less @ real @ X2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z7 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_3586_set__bit__eq,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N3: nat,K3: int] :
( plus_plus @ int @ K3
@ ( times_times @ int
@ ( zero_neq_one_of_bool @ int
@ ~ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) )
@ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% set_bit_eq
thf(fact_3587_unset__bit__eq,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ K3 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_3588_take__bit__Suc__from__most,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K )
= ( 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 @ K @ N ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_3589_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_3590_modulo__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L )
@ ( minus_minus @ int
@ ( semiring_1_of_nat @ int
@ ( times_times @ nat @ N
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M ) ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_3591_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L2 ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_3592_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_3593_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8646137997579335489_i_l_d @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
thf(fact_3594_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
! [X: nat,Y: nat] :
( ( ( vEBT_V8346862874174094_d_u_p @ X )
= Y )
=> ( ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
thf(fact_3595_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_3596_member__bound__height_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_m_e_m_b_e_r2 @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% member_bound_height'
thf(fact_3597_and_Oidem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ A2 )
= A2 ) ) ).
% and.idem
thf(fact_3598_and_Oleft__idem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) ) ) ).
% and.left_idem
thf(fact_3599_and_Oright__idem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) @ B2 )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) ) ) ).
% and.right_idem
thf(fact_3600_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_3601_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_3602_zero__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_and_eq
thf(fact_3603_and__zero__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_zero_eq
thf(fact_3604_take__bit__and,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) ) ) ).
% take_bit_and
thf(fact_3605_and_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A2 )
= A2 ) ) ).
% and.left_neutral
thf(fact_3606_and_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= A2 ) ) ).
% and.right_neutral
thf(fact_3607_bit_Oconj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= X ) ) ).
% bit.conj_one_right
thf(fact_3608_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_3609_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_3610_and__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% and_nonnegative_int_iff
thf(fact_3611_and__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% and_negative_int_iff
thf(fact_3612_zero__le__sgn__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sgn_sgn @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_le_sgn_iff
thf(fact_3613_sgn__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sgn_sgn @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% sgn_le_0_iff
thf(fact_3614_and__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% and_numerals(8)
thf(fact_3615_and__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( one_one @ A ) ) ) ).
% and_numerals(2)
thf(fact_3616_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_3617_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_3618_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_3619_and__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(6)
thf(fact_3620_and__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(2)
thf(fact_3621_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_3622_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_3623_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(5)
thf(fact_3624_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(1)
thf(fact_3625_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_3626_bit__and__int__iff,axiom,
! [K: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L @ N ) ) ) ).
% bit_and_int_iff
thf(fact_3627_bit__and__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
& ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ).
% bit_and_iff
thf(fact_3628_of__nat__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se5824344872417868541ns_and @ nat @ M @ N ) )
= ( bit_se5824344872417868541ns_and @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_and_eq
thf(fact_3629_and_Oassoc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) @ C2 )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ B2 @ C2 ) ) ) ) ).
% and.assoc
thf(fact_3630_and_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5824344872417868541ns_and @ A )
= ( ^ [A5: A,B5: A] : ( bit_se5824344872417868541ns_and @ A @ B5 @ A5 ) ) ) ) ).
% and.commute
thf(fact_3631_and_Oleft__commute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( bit_se5824344872417868541ns_and @ A @ B2 @ ( bit_se5824344872417868541ns_and @ A @ A2 @ C2 ) )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ B2 @ C2 ) ) ) ) ).
% and.left_commute
thf(fact_3632_of__int__and__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,L: int] :
( ( ring_1_of_int @ A @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
= ( bit_se5824344872417868541ns_and @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) ) ) ).
% of_int_and_eq
thf(fact_3633_bit_Oconj__xor__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y: A,Z3: A,X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344971392196577ns_xor @ A @ Y @ Z3 ) @ X )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se5824344872417868541ns_and @ A @ Y @ X ) @ ( bit_se5824344872417868541ns_and @ A @ Z3 @ X ) ) ) ) ).
% bit.conj_xor_distrib2
thf(fact_3634_bit_Oconj__xor__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_se5824344971392196577ns_xor @ A @ Y @ Z3 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se5824344872417868541ns_and @ A @ X @ Y ) @ ( bit_se5824344872417868541ns_and @ A @ X @ Z3 ) ) ) ) ).
% bit.conj_xor_distrib
thf(fact_3635_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( A2
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
& ( B2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_3636_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_3637_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_3638_AND__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) ) ) ).
% AND_lower
thf(fact_3639_AND__upper1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ X ) ) ).
% AND_upper1
thf(fact_3640_AND__upper2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Y ) ) ).
% AND_upper2
thf(fact_3641_AND__upper1_H,axiom,
! [Y: int,Z3: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ Y @ Z3 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z3 ) ) ) ).
% AND_upper1'
thf(fact_3642_AND__upper2_H,axiom,
! [Y: int,Z3: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ Y @ Z3 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Z3 ) ) ) ).
% AND_upper2'
thf(fact_3643_take__bit__eq__mask,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A5: A] : ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ) ).
% take_bit_eq_mask
thf(fact_3644_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_3645_disjunctive__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [B2: A,A2: A] :
( ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 )
=> ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) )
=> ( ( minus_minus @ A @ A2 @ B2 )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_ri4277139882892585799ns_not @ A @ B2 ) ) ) ) ) ).
% disjunctive_diff
thf(fact_3646_and__less__eq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ K ) ) ).
% and_less_eq
thf(fact_3647_AND__upper1_H_H,axiom,
! [Y: int,Z3: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z3 )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z3 ) ) ) ).
% AND_upper1''
thf(fact_3648_AND__upper2_H_H,axiom,
! [Y: int,Z3: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z3 )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Z3 ) ) ) ).
% AND_upper2''
thf(fact_3649_and__not__numerals_I1_J,axiom,
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ).
% and_not_numerals(1)
thf(fact_3650_even__and__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_and_iff
thf(fact_3651_sgn__real__def,axiom,
( ( sgn_sgn @ real )
= ( ^ [A5: real] :
( if @ real
@ ( A5
= ( zero_zero @ real ) )
@ ( zero_zero @ real )
@ ( if @ real @ ( ord_less @ real @ ( zero_zero @ real ) @ A5 ) @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ) ).
% sgn_real_def
thf(fact_3652_even__and__iff__int,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
= ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
| ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ).
% even_and_iff_int
thf(fact_3653_and__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_not_numerals(2)
thf(fact_3654_and__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(4)
thf(fact_3655_bit__nat__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( nat2 @ K ) @ N )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_nat_iff
thf(fact_3656_and__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( one_one @ A ) )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% and_one_eq
thf(fact_3657_one__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A2 )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% one_and_eq
thf(fact_3658_sgn__power__injE,axiom,
! [A2: real,N: nat,X: real,B2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A2 ) @ ( power_power @ real @ ( abs_abs @ real @ A2 ) @ N ) )
= X )
=> ( ( X
= ( times_times @ real @ ( sgn_sgn @ real @ B2 ) @ ( power_power @ real @ ( abs_abs @ real @ B2 ) @ N ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A2 = B2 ) ) ) ) ).
% sgn_power_injE
thf(fact_3659_and__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(5)
thf(fact_3660_and__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(7)
thf(fact_3661_and__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_not_numerals(3)
thf(fact_3662_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_zero @ A ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_3663_bit__nat__def,axiom,
( ( bit_se5641148757651400278ts_bit @ nat )
= ( ^ [M3: nat,N3: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ M3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% bit_nat_def
thf(fact_3664_and__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(9)
thf(fact_3665_and__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(6)
thf(fact_3666_and__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( 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 @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_3667_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_3668_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X2: rat] :
( the @ int
@ ^ [Z7: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z7 ) @ X2 )
& ( ord_less @ rat @ X2 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z7 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_3669_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_3670_and__int_Oelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y )
=> ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( 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 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( 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 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_3671_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_3672_bot__apply,axiom,
! [C: $tType,D3: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D3 > C ) )
= ( ^ [X2: D3] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_3673_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_3674_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_3675_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_3676_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_3677_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_3678_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_3679_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_3680_Diff__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Diff_empty
thf(fact_3681_empty__Diff,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_3682_Diff__cancel,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_3683_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_3684_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X2: A] : X2 = A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_3685_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A3: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_3686_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_3687_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_3688_insert__Diff__single,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A3 ) ) ).
% insert_Diff_single
thf(fact_3689_subset__Compl__singleton,axiom,
! [A: $tType,A3: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_3690_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_3691_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_3692_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_3693_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_3694_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_3695_power2__i,axiom,
( ( power_power @ complex @ imaginary_unit @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% power2_i
thf(fact_3696_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_3697_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_3698_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_3699_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_3700_sgn__rat__def,axiom,
( ( sgn_sgn @ rat )
= ( ^ [A5: rat] :
( if @ rat
@ ( A5
= ( zero_zero @ rat ) )
@ ( zero_zero @ rat )
@ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A5 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% sgn_rat_def
thf(fact_3701_less__eq__rat__def,axiom,
( ( ord_less_eq @ rat )
= ( ^ [X2: rat,Y4: rat] :
( ( ord_less @ rat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% less_eq_rat_def
thf(fact_3702_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_3703_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_3704_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_3705_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_3706_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_3707_singleton__iff,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_3708_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C2: A,D4: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C2 )
& ( B2 = D4 ) )
| ( ( A2 = D4 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_3709_insert__not__empty,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_3710_singleton__inject,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_3711_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_3712_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( A2
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% bot.not_eq_extremum
thf(fact_3713_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $false ) ) ).
% empty_def
thf(fact_3714_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_3715_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_3716_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_3717_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_3718_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_3719_abs__rat__def,axiom,
( ( abs_abs @ rat )
= ( ^ [A5: rat] : ( if @ rat @ ( ord_less @ rat @ A5 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A5 ) @ A5 ) ) ) ).
% abs_rat_def
thf(fact_3720_obtain__pos__sum,axiom,
! [R3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 )
=> ~ ! [S3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ S3 )
=> ! [T4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ T4 )
=> ( R3
!= ( plus_plus @ rat @ S3 @ T4 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_3721_not__psubset__empty,axiom,
! [A: $tType,A3: set @ A] :
~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_3722_complex__i__not__numeral,axiom,
! [W2: num] :
( imaginary_unit
!= ( numeral_numeral @ complex @ W2 ) ) ).
% complex_i_not_numeral
thf(fact_3723_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A2 = X2 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_3724_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A2 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_3725_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_3726_subset__singletonD,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_3727_subset__singleton__iff,axiom,
! [A: $tType,X5: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ( X5
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_3728_Diff__insert,axiom,
! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_3729_insert__Diff,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_3730_Diff__insert2,axiom,
! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_3731_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_3732_subset__Compl__self__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_3733_complex__i__not__neg__numeral,axiom,
! [W2: num] :
( imaginary_unit
!= ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W2 ) ) ) ).
% complex_i_not_neg_numeral
thf(fact_3734_and__nat__def,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se5824344872417868541ns_and @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% and_nat_def
thf(fact_3735_subset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B3 ) )
= ( ( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_3736_Diff__single__insert,axiom,
! [A: $tType,A3: set @ A,X: A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_3737_Compl__insert,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_3738_imaginary__unit_Ocode,axiom,
( imaginary_unit
= ( complex2 @ ( zero_zero @ real ) @ ( one_one @ real ) ) ) ).
% imaginary_unit.code
thf(fact_3739_Complex__eq__i,axiom,
! [X: real,Y: real] :
( ( ( complex2 @ X @ Y )
= imaginary_unit )
= ( ( X
= ( zero_zero @ real ) )
& ( Y
= ( one_one @ real ) ) ) ) ).
% Complex_eq_i
thf(fact_3740_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_3741_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_3742_psubset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B3 ) )
= ( ( ( member @ A @ X @ B3 )
=> ( ord_less @ ( set @ A ) @ A3 @ B3 ) )
& ( ~ ( member @ A @ X @ B3 )
=> ( ( ( member @ A @ X @ A3 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_3743_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M3: nat,N3: nat] :
( if @ nat
@ ( ( M3
= ( zero_zero @ nat ) )
| ( N3
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_3744_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M3: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_3745_Arg__minus__ii,axiom,
( ( arg @ ( uminus_uminus @ complex @ imaginary_unit ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_minus_ii
thf(fact_3746_csqrt__ii,axiom,
( ( csqrt @ imaginary_unit )
= ( divide_divide @ complex @ ( plus_plus @ complex @ ( one_one @ complex ) @ imaginary_unit ) @ ( real_Vector_of_real @ complex @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% csqrt_ii
thf(fact_3747_Arg__ii,axiom,
( ( arg @ imaginary_unit )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_ii
thf(fact_3748_cis__minus__pi__half,axiom,
( ( cis @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ complex @ imaginary_unit ) ) ).
% cis_minus_pi_half
thf(fact_3749_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_3750_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_3751_cis__zero,axiom,
( ( cis @ ( zero_zero @ real ) )
= ( one_one @ complex ) ) ).
% cis_zero
thf(fact_3752_power2__csqrt,axiom,
! [Z3: complex] :
( ( power_power @ complex @ ( csqrt @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z3 ) ).
% power2_csqrt
thf(fact_3753_cis__pi__half,axiom,
( ( cis @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= imaginary_unit ) ).
% cis_pi_half
thf(fact_3754_cis__2pi,axiom,
( ( cis @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ complex ) ) ).
% cis_2pi
thf(fact_3755_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_3756_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_3757_bot__enat__def,axiom,
( ( bot_bot @ extended_enat )
= ( zero_zero @ extended_enat ) ) ).
% bot_enat_def
thf(fact_3758_DeMoivre,axiom,
! [A2: real,N: nat] :
( ( power_power @ complex @ ( cis @ A2 ) @ N )
= ( cis @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A2 ) ) ) ).
% DeMoivre
thf(fact_3759_cis__Arg__unique,axiom,
! [Z3: complex,X: real] :
( ( ( sgn_sgn @ complex @ Z3 )
= ( cis @ X ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( arg @ Z3 )
= X ) ) ) ) ).
% cis_Arg_unique
thf(fact_3760_Arg__correct,axiom,
! [Z3: complex] :
( ( Z3
!= ( zero_zero @ complex ) )
=> ( ( ( sgn_sgn @ complex @ Z3 )
= ( cis @ ( arg @ Z3 ) ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z3 ) )
& ( ord_less_eq @ real @ ( arg @ Z3 ) @ pi ) ) ) ).
% Arg_correct
thf(fact_3761_Arg__zero,axiom,
( ( arg @ ( zero_zero @ complex ) )
= ( zero_zero @ real ) ) ).
% Arg_zero
thf(fact_3762_of__real__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( real_Vector_of_real @ complex @ ( sqrt @ X ) )
= ( csqrt @ ( real_Vector_of_real @ complex @ X ) ) ) ) ).
% of_real_sqrt
thf(fact_3763_Arg__bounded,axiom,
! [Z3: complex] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z3 ) )
& ( ord_less_eq @ real @ ( arg @ Z3 ) @ pi ) ) ).
% Arg_bounded
thf(fact_3764_Arg__def,axiom,
( arg
= ( ^ [Z7: complex] :
( if @ real
@ ( Z7
= ( zero_zero @ complex ) )
@ ( zero_zero @ real )
@ ( fChoice @ real
@ ^ [A5: real] :
( ( ( sgn_sgn @ complex @ Z7 )
= ( cis @ A5 ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ A5 )
& ( ord_less_eq @ real @ A5 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_3765_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_3766_powr__real__of__int,axiom,
! [X: real,N: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( power_power @ real @ X @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( inverse_inverse @ real @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_3767_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X8: set @ A] :
( the @ A
@ ^ [X2: A] :
( X8
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_3768_inverse__nonzero__iff__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_3769_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_3770_inverse__nonpositive__iff__nonpositive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_3771_inverse__nonnegative__iff__nonnegative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_3772_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_3773_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_3774_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_3775_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_3776_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_3777_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_3778_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_3779_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_3780_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( inverse_inverse @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_3781_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_3782_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_3783_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_3784_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_3785_Ints__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_add
thf(fact_3786_minus__in__Ints__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A] :
( ( member @ A @ ( uminus_uminus @ A @ X ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% minus_in_Ints_iff
thf(fact_3787_Ints__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( uminus_uminus @ A @ A2 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_minus
thf(fact_3788_Ints__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( power_power @ A @ A2 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_power
thf(fact_3789_nonzero__norm__inverse,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_3790_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_3791_inverse__zero__imp__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ).
% inverse_zero_imp_zero
thf(fact_3792_nonzero__inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( A2 = B2 ) ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_3793_nonzero__inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
= A2 ) ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_3794_nonzero__imp__inverse__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
!= ( zero_zero @ A ) ) ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_3795_nonzero__of__real__inverse,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [X: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( real_Vector_of_real @ A @ ( inverse_inverse @ real @ X ) )
= ( inverse_inverse @ A @ ( real_Vector_of_real @ A @ X ) ) ) ) ) ).
% nonzero_of_real_inverse
thf(fact_3796_power__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( inverse_inverse @ A @ A2 ) @ N )
= ( inverse_inverse @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_inverse
thf(fact_3797_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_3798_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_3799_Ints__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( abs_abs @ A @ A2 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_abs
thf(fact_3800_Ints__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_nat
thf(fact_3801_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_3802_verit__sko__forall__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P4: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P4 @ X3 ) )
=> ( ( ! [X8: A] : ( P4 @ X8 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_3803_verit__sko__forall__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
=> ( ( ! [X8: A] : ( P @ X8 ) )
= ( P @ X ) ) ) ).
% verit_sko_forall_indirect
thf(fact_3804_verit__sko__ex__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P4: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P4 @ X3 ) )
=> ( ( ? [X8: A] : ( P4 @ X8 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_3805_verit__sko__ex__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ( ? [X8: A] : ( P @ X8 ) )
= ( P @ X ) ) ) ).
% verit_sko_ex_indirect
thf(fact_3806_verit__sko__forall_H_H,axiom,
! [A: $tType,B3: A,A3: A,P: A > $o] :
( ( B3 = A3 )
=> ( ( ( fChoice @ A @ P )
= A3 )
= ( ( fChoice @ A @ P )
= B3 ) ) ) ).
% verit_sko_forall''
thf(fact_3807_verit__sko__forall_H,axiom,
! [A: $tType,P: A > $o,A3: $o] :
( ( ( P
@ ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
= A3 )
=> ( ( ! [X8: A] : ( P @ X8 ) )
= A3 ) ) ).
% verit_sko_forall'
thf(fact_3808_verit__sko__forall,axiom,
! [A: $tType] :
( ( ^ [P2: A > $o] :
! [X4: A] : ( P2 @ X4 ) )
= ( ^ [P3: A > $o] :
( P3
@ ( fChoice @ A
@ ^ [X2: A] :
~ ( P3 @ X2 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_3809_verit__sko__ex_H,axiom,
! [A: $tType,P: A > $o,A3: $o] :
( ( ( P @ ( fChoice @ A @ P ) )
= A3 )
=> ( ( ? [X8: A] : ( P @ X8 ) )
= A3 ) ) ).
% verit_sko_ex'
thf(fact_3810_Ints__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_diff
thf(fact_3811_real__sqrt__inverse,axiom,
! [X: real] :
( ( sqrt @ ( inverse_inverse @ real @ X ) )
= ( inverse_inverse @ real @ ( sqrt @ X ) ) ) ).
% real_sqrt_inverse
thf(fact_3812_Ints__of__int,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: int] : ( member @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_int
thf(fact_3813_Ints__induct,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q3: A,P: A > $o] :
( ( member @ A @ Q3 @ ( ring_1_Ints @ A ) )
=> ( ! [Z: int] : ( P @ ( ring_1_of_int @ A @ Z ) )
=> ( P @ Q3 ) ) ) ) ).
% Ints_induct
thf(fact_3814_Ints__cases,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q3: A] :
( ( member @ A @ Q3 @ ( ring_1_Ints @ A ) )
=> ~ ! [Z: int] :
( Q3
!= ( ring_1_of_int @ A @ Z ) ) ) ) ).
% Ints_cases
thf(fact_3815_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_3816_norm__inverse__le__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [R3: real,X: A] :
( ( ord_less_eq @ real @ R3 @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X ) ) @ ( inverse_inverse @ real @ R3 ) ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_3817_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_3818_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_3819_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_3820_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_3821_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_3822_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_3823_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_3824_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_3825_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_3826_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_3827_nonzero__inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_3828_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( inverse_inverse @ A @ X ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_3829_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_3830_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: nat,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_3831_nonzero__abs__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A2 ) ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_3832_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: int,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3833_divide__real__def,axiom,
( ( divide_divide @ real )
= ( ^ [X2: real,Y4: real] : ( times_times @ real @ X2 @ ( inverse_inverse @ real @ Y4 ) ) ) ) ).
% divide_real_def
thf(fact_3834_Ints__double__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_3835_exp__fdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( diffs @ A
@ ^ [N3: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N3 ) ) )
= ( ^ [N3: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N3 ) ) ) ) ) ).
% exp_fdiffs
thf(fact_3836_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_3837_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_3838_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_3839_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_3840_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_3841_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_3842_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% one_less_inverse
thf(fact_3843_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_3844_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% inverse_add
thf(fact_3845_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_3846_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ B2 @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_3847_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_3848_inverse__powr,axiom,
! [Y: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( powr @ real @ ( inverse_inverse @ real @ Y ) @ A2 )
= ( inverse_inverse @ real @ ( powr @ real @ Y @ A2 ) ) ) ) ).
% inverse_powr
thf(fact_3849_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_3850_of__int__divide__in__Ints,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: int,A2: int] :
( ( dvd_dvd @ int @ B2 @ A2 )
=> ( member @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A2 ) @ ( ring_1_of_int @ A @ B2 ) ) @ ( ring_1_Ints @ A ) ) ) ) ).
% of_int_divide_in_Ints
thf(fact_3851_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ B2 @ A2 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_3852_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_3853_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_3854_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_3855_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% one_le_inverse
thf(fact_3856_inverse__diff__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_3857_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_3858_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D6: real,E: real] :
( ( ord_less @ real @ D6 @ E )
=> ( ( P @ D6 )
=> ( P @ E ) ) )
=> ( ! [N2: nat] : ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_3859_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D6: real,E: real] :
( ( ord_less @ real @ D6 @ E )
=> ( ( P @ D6 )
=> ( P @ E ) ) )
=> ( ! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_3860_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
= ( ? [N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) )
& ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_3861_sqrt__divide__self__eq,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( divide_divide @ real @ ( sqrt @ X ) @ X )
= ( inverse_inverse @ real @ ( sqrt @ X ) ) ) ) ).
% sqrt_divide_self_eq
thf(fact_3862_ln__inverse,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_inverse
thf(fact_3863_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_3864_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_3865_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_3866_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
=> ( ( X = Y )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_3867_summable__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N3 ) ) @ ( power_power @ A @ X @ N3 ) ) ) ) ).
% summable_exp
thf(fact_3868_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_3869_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_3870_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N @ M ) )
= ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_3871_log__inverse,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A2 @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( log @ A2 @ X ) ) ) ) ) ) ).
% log_inverse
thf(fact_3872_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_3873_exp__plus__inverse__exp,axiom,
! [X: real] : ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) ) ).
% exp_plus_inverse_exp
thf(fact_3874_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: A] :
( ( ( archimedean_frac @ A @ X )
= A2 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A2 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_3875_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A2: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 )
=> ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A2 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A2 @ B2 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_3876_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A2: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 )
=> ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A2 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A2 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_3877_plus__inverse__ge__2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X @ ( inverse_inverse @ real @ X ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_3878_real__inv__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( inverse_inverse @ real @ X ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_3879_tan__cot,axiom,
! [X: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) )
= ( inverse_inverse @ real @ ( tan @ real @ X ) ) ) ).
% tan_cot
thf(fact_3880_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_3881_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_3882_real__le__x__sinh,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_x_sinh
thf(fact_3883_real__le__abs__sinh,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( abs_abs @ real @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_abs_sinh
thf(fact_3884_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_3885_some__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ( fChoice @ A @ P )
= A2 ) ) ) ).
% some_equality
thf(fact_3886_some__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ^ [Y4: A] : Y4 = X )
= X ) ).
% some_eq_trivial
thf(fact_3887_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_3888_exp__first__two__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X2 )
@ ( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X2 @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_3889_scaleR__cancel__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A,B2: real] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
= ( ( A2 = B2 )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_cancel_right
thf(fact_3890_scaleR__zero__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_right
thf(fact_3891_scaleR__cancel__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A,Y: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) )
= ( ( X = Y )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_cancel_left
thf(fact_3892_scaleR__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ real ) )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_eq_0_iff
thf(fact_3893_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_3894_scaleR__power,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: real,Y: A,N: nat] :
( ( power_power @ A @ ( real_V8093663219630862766scaleR @ A @ X @ Y ) @ N )
= ( real_V8093663219630862766scaleR @ A @ ( power_power @ real @ X @ N ) @ ( power_power @ A @ Y @ N ) ) ) ) ).
% scaleR_power
thf(fact_3895_scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U2: num,W2: num,A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ U2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ A2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ ( numeral_numeral @ real @ U2 ) @ ( numeral_numeral @ real @ W2 ) ) @ A2 ) ) ) ).
% scaleR_times
thf(fact_3896_inverse__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [V2: num,W2: num,A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ A2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ W2 ) @ ( numeral_numeral @ real @ V2 ) ) @ A2 ) ) ) ).
% inverse_scaleR_times
thf(fact_3897_fraction__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U2: num,V2: num,W2: num,A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ U2 ) @ ( numeral_numeral @ real @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ A2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ U2 ) @ ( numeral_numeral @ real @ W2 ) ) @ ( numeral_numeral @ real @ V2 ) ) @ A2 ) ) ) ).
% fraction_scaleR_times
thf(fact_3898_scaleR__half__double,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ A2 @ A2 ) )
= A2 ) ) ).
% scaleR_half_double
thf(fact_3899_scaleR__left__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A,Y: A] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) )
=> ( X = Y ) ) ) ) ).
% scaleR_left_imp_eq
thf(fact_3900_scaleR__right__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,A2: real,B2: real] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
=> ( A2 = B2 ) ) ) ) ).
% scaleR_right_imp_eq
thf(fact_3901_summable__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: nat > B,R3: real] :
( ( summable @ B @ X5 )
=> ( summable @ B
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ B @ R3 @ ( X5 @ N3 ) ) ) ) ) ).
% summable_scaleR_right
thf(fact_3902_sums__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: nat > B,A2: B,R3: real] :
( ( sums @ B @ X5 @ A2 )
=> ( sums @ B
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ B @ R3 @ ( X5 @ N3 ) )
@ ( real_V8093663219630862766scaleR @ B @ R3 @ A2 ) ) ) ) ).
% sums_scaleR_right
thf(fact_3903_suminf__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: nat > B,R3: real] :
( ( summable @ B @ X5 )
=> ( ( real_V8093663219630862766scaleR @ B @ R3 @ ( suminf @ B @ X5 ) )
= ( suminf @ B
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ B @ R3 @ ( X5 @ N3 ) ) ) ) ) ) ).
% suminf_scaleR_right
thf(fact_3904_summable__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: nat > real,X: B] :
( ( summable @ real @ X5 )
=> ( summable @ B
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ B @ ( X5 @ N3 ) @ X ) ) ) ) ).
% summable_scaleR_left
thf(fact_3905_sums__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: nat > real,A2: real,X: B] :
( ( sums @ real @ X5 @ A2 )
=> ( sums @ B
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ B @ ( X5 @ N3 ) @ X )
@ ( real_V8093663219630862766scaleR @ B @ A2 @ X ) ) ) ) ).
% sums_scaleR_left
thf(fact_3906_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_3907_scaleR__right__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: real,A2: real,C2: A] :
( ( ord_less_eq @ real @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ C2 ) ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_3908_scaleR__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_3909_scaleR__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_3910_scaleR__le__cancel__left,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_3911_scaleR__left__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: A,A2: A,C2: real] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ real @ C2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_3912_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,Y: A,A2: real] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_3913_eq__vector__fraction__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,U2: real,V2: real,A2: A] :
( ( X
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U2 @ V2 ) @ A2 ) )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ V2 @ X )
= ( real_V8093663219630862766scaleR @ A @ U2 @ A2 ) ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_3914_vector__fraction__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U2: real,V2: real,A2: A,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U2 @ V2 ) @ A2 )
= X )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ U2 @ A2 )
= ( real_V8093663219630862766scaleR @ A @ V2 @ X ) ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_3915_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E2: A,C2: A,B2: real,D4: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E2 ) @ D4 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ E2 ) @ C2 ) @ D4 ) ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_3916_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E2: A,C2: A,B2: real,D4: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E2 ) @ D4 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B2 @ A2 ) @ E2 ) @ D4 ) ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_3917_suminf__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X5: nat > real,X: B] :
( ( summable @ real @ X5 )
=> ( ( real_V8093663219630862766scaleR @ B @ ( suminf @ real @ X5 ) @ X )
= ( suminf @ B
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ B @ ( X5 @ N3 ) @ X ) ) ) ) ) ).
% suminf_scaleR_left
thf(fact_3918_scaleR__le__0__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_le_0_iff
thf(fact_3919_zero__le__scaleR__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% zero_le_scaleR_iff
thf(fact_3920_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X: A,Y: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_3921_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,C2: A,D4: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D4 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ D4 ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_3922_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_3923_split__scaleR__pos__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ).
% split_scaleR_pos_le
thf(fact_3924_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_3925_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_3926_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_3927_scaleR__nonpos__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_3928_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,A2: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ real @ A2 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ X ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_3929_scaleR__2,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X )
= ( plus_plus @ A @ X @ X ) ) ) ).
% scaleR_2
thf(fact_3930_real__vector__affinity__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,X: A,C2: A,Y: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X ) @ C2 )
= Y )
= ( X
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_3931_real__vector__eq__affinity,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,Y: A,X: A,C2: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( Y
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X ) @ C2 ) )
= ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) )
= X ) ) ) ) ).
% real_vector_eq_affinity
thf(fact_3932_neg__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_le_divideR_eq
thf(fact_3933_neg__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% neg_divideR_le_eq
thf(fact_3934_pos__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% pos_le_divideR_eq
thf(fact_3935_pos__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_divideR_le_eq
thf(fact_3936_neg__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_3937_neg__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% neg_divideR_less_eq
thf(fact_3938_pos__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% pos_less_divideR_eq
thf(fact_3939_pos__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_3940_nonzero__inverse__scaleR__distrib,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A2: real,X: A] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) )
= ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ A2 ) @ ( inverse_inverse @ A @ X ) ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_3941_summable__exp__generic,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ N3 ) ) ) ) ).
% summable_exp_generic
thf(fact_3942_sin__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X @ N3 ) )
@ ( sin @ A @ X ) ) ) ).
% sin_converges
thf(fact_3943_sin__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ) ).
% sin_def
thf(fact_3944_cos__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X @ N3 ) )
@ ( cos @ A @ X ) ) ) ).
% cos_converges
thf(fact_3945_cos__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ) ).
% cos_def
thf(fact_3946_summable__norm__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ).
% summable_norm_sin
thf(fact_3947_summable__norm__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ).
% summable_norm_cos
thf(fact_3948_neg__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_le_eq
thf(fact_3949_neg__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_le_minus_divideR_eq
thf(fact_3950_pos__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_minus_divideR_le_eq
thf(fact_3951_pos__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divideR_eq
thf(fact_3952_neg__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_3953_neg__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_3954_pos__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_3955_pos__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_3956_some1__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( ( P @ A2 )
=> ( ( fChoice @ A @ P )
= A2 ) ) ) ).
% some1_equality
thf(fact_3957_some__eq__ex,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X8: A] : ( P @ X8 ) ) ) ).
% some_eq_ex
thf(fact_3958_someI2__bex,axiom,
! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
( ? [X6: A] :
( ( member @ A @ X6 @ A3 )
& ( P @ X6 ) )
=> ( ! [X3: A] :
( ( ( member @ A @ X3 @ A3 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_3959_someI2__ex,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2_ex
thf(fact_3960_someI__ex,axiom,
! [A: $tType,P: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( P @ ( fChoice @ A @ P ) ) ) ).
% someI_ex
thf(fact_3961_someI2,axiom,
! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2
thf(fact_3962_exp__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ N3 ) )
@ ( exp @ A @ X ) ) ) ).
% exp_converges
thf(fact_3963_exp__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ) ).
% exp_def
thf(fact_3964_summable__norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N3: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ N3 ) ) ) ) ) ).
% summable_norm_exp
thf(fact_3965_sin__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N3 ) ) )
@ ( sin @ A @ X ) ) ) ).
% sin_minus_converges
thf(fact_3966_cos__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N3 ) )
@ ( cos @ A @ X ) ) ) ).
% cos_minus_converges
thf(fact_3967_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_1: A] : ( P @ ( zero_zero @ nat ) @ X_1 )
=> ( ! [X3: A,N2: nat] :
( ( P @ N2 @ X3 )
=> ? [Y2: A] :
( ( P @ ( suc @ N2 ) @ Y2 )
& ( Q @ N2 @ X3 @ Y2 ) ) )
=> ? [F5: nat > A] :
! [N9: nat] :
( ( P @ N9 @ ( F5 @ N9 ) )
& ( Q @ N9 @ ( F5 @ N9 ) @ ( F5 @ ( suc @ N9 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_3968_complex__inverse,axiom,
! [A2: real,B2: real] :
( ( inverse_inverse @ complex @ ( complex2 @ A2 @ B2 ) )
= ( complex2 @ ( divide_divide @ real @ A2 @ ( plus_plus @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( uminus_uminus @ real @ B2 ) @ ( plus_plus @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_inverse
thf(fact_3969_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N3 ) ) ) @ ( power_power @ A @ X2 @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_3970_some__in__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
@ A3 )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_3971_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( sinh @ A @ X ) ) ) ).
% sinh_converges
thf(fact_3972_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X @ N3 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X ) ) ) ).
% cosh_converges
thf(fact_3973_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) ) @ ( power_power @ A @ X @ N3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N3 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ).
% sin_x_sin_y
thf(fact_3974_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 ) @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X @ N3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N3 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_3975_sinh__real__zero__iff,axiom,
! [X: real] :
( ( ( sinh @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% sinh_real_zero_iff
thf(fact_3976_sinh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( sinh @ real @ X ) @ ( sinh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% sinh_real_less_iff
thf(fact_3977_sinh__real__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( sinh @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ).
% sinh_real_le_iff
thf(fact_3978_sinh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sinh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% sinh_real_neg_iff
thf(fact_3979_sinh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% sinh_real_pos_iff
thf(fact_3980_sinh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% sinh_real_nonneg_iff
thf(fact_3981_sinh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% sinh_real_nonpos_iff
thf(fact_3982_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
= ( ord_less_eq @ A @ I @ K ) ) ) ).
% atMost_iff
thf(fact_3983_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_3984_Ints__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ B )
=> ! [A3: set @ A,F2: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ B @ ( F2 @ X3 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_sum
thf(fact_3985_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F2: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F2 @ X2 ) )
@ A3 ) ) ) ).
% of_nat_sum
thf(fact_3986_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F2: B > int,A3: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( ring_1_of_int @ A @ ( F2 @ X2 ) )
@ A3 ) ) ) ).
% of_int_sum
thf(fact_3987_of__real__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [F2: B > real,S: set @ B] :
( ( real_Vector_of_real @ A @ ( groups7311177749621191930dd_sum @ B @ real @ F2 @ S ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( real_Vector_of_real @ A @ ( F2 @ X2 ) )
@ S ) ) ) ).
% of_real_sum
thf(fact_3988_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_3989_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_3990_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_3991_scaleR__left_Osum,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [G: C > real,A3: set @ C,X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( groups7311177749621191930dd_sum @ C @ real @ G @ A3 ) @ X )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ A @ ( G @ X2 ) @ X )
@ A3 ) ) ) ).
% scaleR_left.sum
thf(fact_3992_scaleR__sum__left,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [F2: C > real,A3: set @ C,X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( groups7311177749621191930dd_sum @ C @ real @ F2 @ A3 ) @ X )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [A5: C] : ( real_V8093663219630862766scaleR @ A @ ( F2 @ A5 ) @ X )
@ A3 ) ) ) ).
% scaleR_sum_left
thf(fact_3993_cosh__real__nonzero,axiom,
! [X: real] :
( ( cosh @ real @ X )
!= ( zero_zero @ real ) ) ).
% cosh_real_nonzero
thf(fact_3994_sum__choose__upper,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ K3 @ M )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_3995_sinh__less__cosh__real,axiom,
! [X: real] : ( ord_less @ real @ ( sinh @ real @ X ) @ ( cosh @ real @ X ) ) ).
% sinh_less_cosh_real
thf(fact_3996_sum__norm__le,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S2: set @ B,F2: B > A,G: B > real] :
( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X3 ) ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S2 ) ) @ ( groups7311177749621191930dd_sum @ B @ real @ G @ S2 ) ) ) ) ).
% sum_norm_le
thf(fact_3997_norm__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,A3: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) )
@ ( groups7311177749621191930dd_sum @ B @ real
@ ^ [I4: B] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ I4 ) )
@ A3 ) ) ) ).
% norm_sum
thf(fact_3998_sinh__le__cosh__real,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( cosh @ real @ X ) ) ).
% sinh_le_cosh_real
thf(fact_3999_mod__sum__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F2: B > A,A2: A,A3: set @ B] :
( ( modulo_modulo @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I4: B] : ( modulo_modulo @ A @ ( F2 @ I4 ) @ A2 )
@ A3 )
@ A2 )
= ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ A2 ) ) ) ).
% mod_sum_eq
thf(fact_4000_scaleR__sum__right,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,F2: C > A,A3: set @ C] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( groups7311177749621191930dd_sum @ C @ A @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ A @ A2 @ ( F2 @ X2 ) )
@ A3 ) ) ) ).
% scaleR_sum_right
thf(fact_4001_scaleR__right_Osum,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,G: C > A,A3: set @ C] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( groups7311177749621191930dd_sum @ C @ A @ G @ A3 ) )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ A @ A2 @ ( G @ X2 ) )
@ A3 ) ) ) ).
% scaleR_right.sum
thf(fact_4002_summable__sum,axiom,
! [I5: $tType,A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I6: set @ I5,F2: I5 > nat > A] :
( ! [I2: I5] :
( ( member @ I5 @ I2 @ I6 )
=> ( summable @ A @ ( F2 @ I2 ) ) )
=> ( summable @ A
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ I5 @ A
@ ^ [I4: I5] : ( F2 @ I4 @ N3 )
@ I6 ) ) ) ) ).
% summable_sum
thf(fact_4003_sums__sum,axiom,
! [A: $tType,I5: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I6: set @ I5,F2: I5 > nat > A,X: I5 > A] :
( ! [I2: I5] :
( ( member @ I5 @ I2 @ I6 )
=> ( sums @ A @ ( F2 @ I2 ) @ ( X @ I2 ) ) )
=> ( sums @ A
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ I5 @ A
@ ^ [I4: I5] : ( F2 @ I4 @ N3 )
@ I6 )
@ ( groups7311177749621191930dd_sum @ I5 @ A @ X @ I6 ) ) ) ) ).
% sums_sum
thf(fact_4004_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_4005_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F2 @ I4 ) @ ( F2 @ ( suc @ I4 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_4006_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,D4: nat > A] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( D4 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
=> ( ( C2 @ I4 )
= ( D4 @ I4 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_4007_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: nat > A,B3: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_ord_atMost @ nat @ N2 ) ) @ B3 )
=> ( summable @ A @ A2 ) ) ) ) ).
% bounded_imp_summable
thf(fact_4008_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ U ) ) ) ) ) ).
% atMost_def
thf(fact_4009_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_4010_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_4011_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% choose_rising_sum(1)
thf(fact_4012_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C2: nat > A,N: nat,K: nat] :
( ! [W: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ W @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( C2 @ K )
= ( zero_zero @ A ) ) ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_4013_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
=> ( ( C2 @ I4 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_4014_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_4015_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus @ nat @ N @ K3 ) @ ( minus_minus @ nat @ M @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( suc @ N ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_4016_vandermonde,axiom,
! [M: nat,N: nat,R3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus @ nat @ R3 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R3 ) )
= ( binomial @ ( plus_plus @ nat @ M @ N ) @ R3 ) ) ).
% vandermonde
thf(fact_4017_cosh__real__pos,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_pos
thf(fact_4018_cosh__real__nonneg,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_nonneg
thf(fact_4019_cosh__real__nonneg__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_le_iff
thf(fact_4020_cosh__real__nonpos__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less_eq @ real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_le_iff
thf(fact_4021_cosh__real__ge__1,axiom,
! [X: real] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_ge_1
thf(fact_4022_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_4023_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_4024_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_4025_suminf__sum,axiom,
! [A: $tType,I5: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I6: set @ I5,F2: I5 > nat > A] :
( ! [I2: I5] :
( ( member @ I5 @ I2 @ I6 )
=> ( summable @ A @ ( F2 @ I2 ) ) )
=> ( ( suminf @ A
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ I5 @ A
@ ^ [I4: I5] : ( F2 @ I4 @ N3 )
@ I6 ) )
= ( groups7311177749621191930dd_sum @ I5 @ A
@ ^ [I4: I5] : ( suminf @ A @ ( F2 @ I4 ) )
@ I6 ) ) ) ) ).
% suminf_sum
thf(fact_4026_choose__row__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ N ) @ ( set_ord_atMost @ nat @ N ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% choose_row_sum
thf(fact_4027_binomial,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K3 ) ) @ ( power_power @ nat @ A2 @ K3 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ).
% binomial
thf(fact_4028_summable__Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( summable @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ).
% summable_Cauchy_product
thf(fact_4029_Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( ( times_times @ A @ ( suminf @ A @ A2 ) @ ( suminf @ A @ B2 ) )
= ( suminf @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ) ).
% Cauchy_product
thf(fact_4030_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_4031_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M: nat,A2: nat > A,N: nat,B2: nat > A,X: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ M @ I2 )
=> ( ( A2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( times_times @ A @ ( B2 @ J3 ) @ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [R5: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ A @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_4032_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_4033_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( power_power @ A @ A2 @ K3 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_4034_polynomial__product__nat,axiom,
! [M: nat,A2: nat > nat,N: nat,B2: nat > nat,X: nat] :
( ! [I2: nat] :
( ( ord_less @ nat @ M @ I2 )
=> ( ( A2 @ I2 )
= ( zero_zero @ nat ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( A2 @ I4 ) @ ( power_power @ nat @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( times_times @ nat @ ( B2 @ J3 ) @ ( power_power @ nat @ X @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [R5: nat] :
( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ nat @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_4035_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_4036_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A2 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_4037_cosh__real__strict__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_4038_cosh__real__nonneg__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_4039_cosh__real__nonpos__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_4040_Cauchy__product__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( sums @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( times_times @ A @ ( suminf @ A @ A2 ) @ ( suminf @ A @ B2 ) ) ) ) ) ) ).
% Cauchy_product_sums
thf(fact_4041_cosh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% cosh_square_eq
thf(fact_4042_hyperbolic__pythagoras,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% hyperbolic_pythagoras
thf(fact_4043_sinh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% sinh_square_eq
thf(fact_4044_atMost__nat__numeral,axiom,
! [K: num] :
( ( set_ord_atMost @ nat @ ( numeral_numeral @ nat @ K ) )
= ( insert @ nat @ ( numeral_numeral @ nat @ K ) @ ( set_ord_atMost @ nat @ ( pred_numeral @ K ) ) ) ) ).
% atMost_nat_numeral
thf(fact_4045_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,I6: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M @ I4 ) )
@ I6 )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I6 ) ) ) ) ).
% sum_power_add
thf(fact_4046_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P5: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
=> ( ( ord_less_eq @ nat @ K @ P5 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P5 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_4047_arcosh__cosh__real,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( arcosh @ real @ ( cosh @ real @ X ) )
= X ) ) ).
% arcosh_cosh_real
thf(fact_4048_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_4049_exp__series__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I4 ) ) @ ( power_power @ A @ X @ I4 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ I4 ) ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_4050_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_4051_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,Z3: A,A2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( ( power_power @ A @ Z3 @ N )
= A2 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] :
( times_times @ A
@ ( if @ A
@ ( I4
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A2 )
@ ( if @ A @ ( I4 = N ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_4052_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_4053_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_4054_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K3 ) ) @ K3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_4055_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A2 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_4056_binomial__r__part__sum,axiom,
! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% binomial_r_part_sum
thf(fact_4057_choose__linear__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ I4 @ ( binomial @ N @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% choose_linear_sum
thf(fact_4058_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_4059_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E2: real,C2: nat > A,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M8: real] :
! [Z4: A] :
( ( ord_less_eq @ real @ M8 @ ( real_V7770717601297561774m_norm @ A @ Z4 ) )
=> ( ord_less_eq @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
@ ( times_times @ real @ E2 @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z4 ) @ ( suc @ N ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_4060_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% gbinomial_r_part_sum
thf(fact_4061_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A2 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_4062_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_4063_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_4064_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_4065_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q6: nat] : ( ord_less @ nat @ Q6 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_4066_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_4067_cosh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cosh @ A )
= ( ^ [Z7: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z7 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z7 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_field_def
thf(fact_4068_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q6: nat] : ( ord_less @ nat @ Q6 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_4069_sinh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sinh @ A )
= ( ^ [Z7: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z7 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z7 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sinh_field_def
thf(fact_4070_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_4071_cosh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A )
= ( ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ) ).
% cosh_def
thf(fact_4072_cosh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( cosh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( plus_plus @ real @ X @ ( inverse_inverse @ real @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% cosh_ln_real
thf(fact_4073_sinh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sinh @ A )
= ( ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ) ).
% sinh_def
thf(fact_4074_sinh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( sinh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( minus_minus @ real @ X @ ( inverse_inverse @ real @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% sinh_ln_real
thf(fact_4075_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N3 ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X @ N3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N3 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ).
% cos_x_cos_y
thf(fact_4076_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A3: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F2 @ I4 ) )
@ A3 ) ) ) ).
% sum_abs_ge_zero
thf(fact_4077_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A3: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F2 @ I4 ) )
@ A3 ) ) ) ).
% sum_abs
thf(fact_4078_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_4079_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,X: A > B,A2: A > B,B2: B,Delta: B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I6 )
= ( one_one @ B ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A2 @ I2 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( times_times @ B @ ( A2 @ I4 ) @ ( X @ I4 ) )
@ I6 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_4080_of__nat__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( ^ [N3: nat] : N3 ) ) ).
% of_nat_id
thf(fact_4081_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu: B] : ( zero_zero @ A )
@ A3 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_4082_abs__sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A3: set @ A] :
( ( abs_abs @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A5: A] : ( abs_abs @ B @ ( F2 @ A5 ) )
@ A3 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A5: A] : ( abs_abs @ B @ ( F2 @ A5 ) )
@ A3 ) ) ) ).
% abs_sum_abs
thf(fact_4083_Complex__sum_H,axiom,
! [A: $tType,F2: A > real,S: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ complex
@ ^ [X2: A] : ( complex2 @ ( F2 @ X2 ) @ ( zero_zero @ real ) )
@ S )
= ( complex2 @ ( groups7311177749621191930dd_sum @ A @ real @ F2 @ S ) @ ( zero_zero @ real ) ) ) ).
% Complex_sum'
thf(fact_4084_int__sum,axiom,
! [B: $tType,F2: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F2 @ X2 ) )
@ A3 ) ) ).
% int_sum
thf(fact_4085_sum__subtractf__nat,axiom,
! [A: $tType,A3: set @ A,G: A > nat,F2: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ nat @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( minus_minus @ nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A3 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A3 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_4086_sum__SucD,axiom,
! [A: $tType,F2: A > nat,A3: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 )
= ( suc @ N ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_4087_sum__diff1__nat,axiom,
! [A: $tType,A2: A,A3: set @ A,F2: A > nat] :
( ( ( member @ A @ A2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) )
& ( ~ ( member @ A @ A2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) ) ) ) ).
% sum_diff1_nat
thf(fact_4088_sum__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X2: complex] : X2
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= C2 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_4089_sum_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > C > A,B3: set @ C,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I4: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ I4 ) @ B3 )
@ A3 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [J3: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [I4: B] : ( G @ I4 @ J3 )
@ A3 )
@ B3 ) ) ) ).
% sum.swap
thf(fact_4090_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X2: complex] : X2
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_4091_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,A3: set @ B] :
( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
!= ( zero_zero @ A ) )
=> ~ ! [A6: B] :
( ( member @ B @ A6 @ A3 )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4092_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_4093_dvd__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: set @ B,D4: A,F2: B > A] :
( ! [A6: B] :
( ( member @ B @ A6 @ A3 )
=> ( dvd_dvd @ A @ D4 @ ( F2 @ A6 ) ) )
=> ( dvd_dvd @ A @ D4 @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ).
% dvd_sum
thf(fact_4094_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K6: set @ B,F2: B > A,G: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ K6 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ K6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K6 ) ) ) ) ).
% sum_mono
thf(fact_4095_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F2: A > B,A3: set @ A,G: C > B,B3: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B3 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F2 @ I4 ) @ ( G @ J3 ) )
@ B3 )
@ A3 ) ) ) ).
% sum_product
thf(fact_4096_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F2: B > A,A3: set @ B,R3: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ R3 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N3: B] : ( times_times @ A @ ( F2 @ N3 ) @ R3 )
@ A3 ) ) ) ).
% sum_distrib_right
thf(fact_4097_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R3: A,F2: B > A,A3: set @ B] :
( ( times_times @ A @ R3 @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N3: B] : ( times_times @ A @ R3 @ ( F2 @ N3 ) )
@ A3 ) ) ) ).
% sum_distrib_left
thf(fact_4098_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,H2: B > A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ A3 ) ) ) ) ).
% sum.distrib
thf(fact_4099_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,G: B > A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A3 )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum_subtractf
thf(fact_4100_sum__divide__distrib,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F2: B > A,A3: set @ B,R3: A] :
( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ R3 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N3: B] : ( divide_divide @ A @ ( F2 @ N3 ) @ R3 )
@ A3 ) ) ) ).
% sum_divide_distrib
thf(fact_4101_sum__negf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( F2 @ X2 ) )
@ A3 )
= ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ).
% sum_negf
thf(fact_4102_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ).
% sum_nonneg
thf(fact_4103_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_4104_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ nat,F2: nat > A,G: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( ! [X3: nat] :
( ( member @ nat @ ( suc @ X3 ) @ A3 )
=> ( ( F2 @ ( suc @ X3 ) )
= ( G @ ( suc @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A3 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ A3 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_4105_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ? [T4: real] :
( ( ord_less @ real @ X @ T4 )
& ( ord_less @ real @ T4 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( cos_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T4 @ ( 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_4106_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less @ real @ T4 @ X )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( cos_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T4 @ ( 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_4107_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less @ real @ T4 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T4 @ ( 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_4108_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T4 @ ( 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_4109_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
= ( ord_less @ A @ I @ K ) ) ) ).
% lessThan_iff
thf(fact_4110_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_4111_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_4112_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A] :
( ( minus_minus @ ( set @ A ) @ ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) )
= ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_4113_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( cos_coeff @ M3 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M3 ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( one_one @ real ) ) ).
% sumr_cos_zero_one
thf(fact_4114_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ U ) ) ) ) ) ).
% lessThan_def
thf(fact_4115_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
= ( ord_less @ A @ M @ N ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_4116_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_4117_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_4118_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A2 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_4119_lessThan__nat__numeral,axiom,
! [K: num] :
( ( set_ord_lessThan @ nat @ ( numeral_numeral @ nat @ K ) )
= ( insert @ nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan @ nat @ ( pred_numeral @ K ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_4120_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_4121_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N: A] :
( ! [X3: A] : ( ord_less_eq @ nat @ ( Q @ X3 ) @ ( P @ X3 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( minus_minus @ nat @ ( P @ X2 ) @ ( Q @ X2 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_4122_suminf__le__const,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X: A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ X )
=> ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ X ) ) ) ) ).
% suminf_le_const
thf(fact_4123_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_4124_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_4125_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_4126_sumr__diff__mult__const2,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [F2: nat > A,N: nat,R3: A] :
( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ R3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F2 @ I4 ) @ R3 )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sumr_diff_mult_const2
thf(fact_4127_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ X )
=> ( summable @ A @ F2 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_4128_sums__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N: nat,S: A] :
( ( sums @ A
@ ^ [I4: nat] : ( F2 @ ( plus_plus @ nat @ I4 @ N ) )
@ S )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_iff_shift
thf(fact_4129_sums__iff__shift_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N: nat,S: A] :
( ( sums @ A
@ ^ [I4: nat] : ( F2 @ ( plus_plus @ nat @ I4 @ N ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) )
= ( sums @ A @ F2 @ S ) ) ) ).
% sums_iff_shift'
thf(fact_4130_sums__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A,N: nat] :
( ( sums @ A @ F2 @ S )
=> ( sums @ A
@ ^ [I4: nat] : ( F2 @ ( plus_plus @ nat @ I4 @ N ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_4131_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_4132_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_4133_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_4134_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_4135_suminf__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A @ F2 )
= ( plus_plus @ A
@ ( suminf @ A
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ K ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_4136_suminf__minus__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ K ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_4137_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N: nat] :
( ( summable @ A @ F2 )
=> ( ! [M4: nat] :
( ( ord_less_eq @ nat @ N @ M4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ M4 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_4138_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_4139_lemma__termdiff1,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [Z3: A,H2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ ( minus_minus @ nat @ M @ P6 ) ) @ ( power_power @ A @ Z3 @ P6 ) ) @ ( power_power @ A @ Z3 @ M ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ Z3 @ P6 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ ( minus_minus @ nat @ M @ P6 ) ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ M @ P6 ) ) ) )
@ ( set_ord_lessThan @ nat @ M ) ) ) ) ).
% lemma_termdiff1
thf(fact_4140_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X @ P6 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_4141_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_4142_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,A2: A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ A2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ~ ! [B6: nat > A] :
~ ! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( B6 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_4143_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,N: nat,A2: A] :
? [B6: nat > A] :
! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A
@ ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( B6 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ A2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% polyfun_linear_factor
thf(fact_4144_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,F2: nat > A,K6: A,K: nat] :
( ! [P7: nat] :
( ( ord_less @ nat @ P7 @ N )
=> ( ord_less_eq @ A @ ( F2 @ P7 ) @ K6 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K6 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ K6 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_4145_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N: nat,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [M4: nat] :
( ( ord_less_eq @ nat @ N @ M4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ M4 ) ) )
=> ( ( ord_less_eq @ nat @ N @ I )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_4146_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_4147_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: real,N: nat,Diff: nat > A > real] :
( ( X
= ( zero_zero @ real ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_4148_Maclaurin__lemma,axiom,
! [H2: real,F2: real > real,J: nat > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ? [B7: real] :
( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J @ M3 ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ H2 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ B7 @ ( divide_divide @ real @ ( power_power @ real @ H2 @ N ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_4149_sum__split__even__odd,axiom,
! [F2: nat > real,G: nat > real,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( F2 @ I4 ) @ ( G @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( F2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( G @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( one_one @ nat ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_4150_Maclaurin__exp__le,axiom,
! [X: real,N: nat] :
? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M3 ) @ ( semiring_char_0_fact @ real @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_4151_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ Y @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( A2 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J3 @ K3 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y @ K3 ) ) @ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ J3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_4152_exp__first__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [K: nat] :
( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N3 ) ) @ ( power_power @ A @ X2 @ N3 ) )
@ ( set_ord_lessThan @ nat @ K ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N3 @ K ) ) ) @ ( power_power @ A @ X2 @ ( plus_plus @ nat @ N3 @ K ) ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_4153_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
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( 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_4154_sum__pos__lt__pair,axiom,
! [F2: nat > real,K: nat] :
( ( summable @ real @ F2 )
=> ( ! [D6: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F2 @ ( plus_plus @ nat @ K @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D6 ) ) ) @ ( F2 @ ( plus_plus @ nat @ K @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D6 ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F2 @ ( set_ord_lessThan @ nat @ K ) ) @ ( suminf @ real @ F2 ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_4155_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T4 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M3 ) @ ( semiring_char_0_fact @ real @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_4156_lemma__termdiff2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [H2: A,Z3: A,N: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ N ) @ ( power_power @ A @ Z3 @ N ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
= ( times_times @ A @ H2
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q6: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ Q6 ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q6 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_4157_Maclaurin__sin__expansion,axiom,
! [X: real,N: nat] :
? [T4: real] :
( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T4 @ ( 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_4158_Maclaurin__sin__expansion2,axiom,
! [X: real,N: nat] :
? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T4 @ ( 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_4159_Maclaurin__cos__expansion,axiom,
! [X: real,N: nat] :
? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( cos_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T4 @ ( 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_4160_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
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= ( one_one @ complex ) ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_4161_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M: nat,X: A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_4162_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R3: A,M: 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 ) @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R3 @ ( suc @ M ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_4163_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_4164_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U2: A] :
( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U2 ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U2 ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_4165_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H2: A,L4: A,H3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L @ H2 )
= ( set_or1337092689740270186AtMost @ A @ L4 @ H3 ) )
= ( ( ( L = L4 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq @ A @ L @ H2 )
& ~ ( ord_less_eq @ A @ L4 @ H3 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4166_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_4167_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_4168_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D4 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D4 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4169_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_4170_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( B2 = C2 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_4171_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
( ( set_or1337092689740270186AtMost @ A @ A2 @ A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_4172_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H2: A,H3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atMost @ A @ H3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H2 )
| ( ord_less_eq @ A @ H2 @ H3 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_4173_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_4174_sum_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [H2: B > C,S2: set @ B,T5: set @ C,G: C > A] :
( ( bij_betw @ B @ C @ H2 @ S2 @ T5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G @ ( H2 @ X2 ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T5 ) ) ) ) ).
% sum.reindex_bij_betw
thf(fact_4175_bij__betwI_H,axiom,
! [A: $tType,B: $tType,X5: set @ A,F2: A > B,Y6: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ X5 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
= ( X3 = Y3 ) ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ( member @ B @ ( F2 @ X3 ) @ Y6 ) )
=> ( ! [Y3: B] :
( ( member @ B @ Y3 @ Y6 )
=> ? [X6: A] :
( ( member @ A @ X6 @ X5 )
& ( Y3
= ( F2 @ X6 ) ) ) )
=> ( bij_betw @ A @ B @ F2 @ X5 @ Y6 ) ) ) ) ).
% bij_betwI'
thf(fact_4176_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_4177_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L4: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H3 ) ) ) ).
% not_Iic_le_Icc
thf(fact_4178_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_4179_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_4180_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_4181_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_4182_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_4183_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D4 ) )
= ( ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D4 )
& ( ( ord_less @ A @ C2 @ A2 )
| ( ord_less @ A @ B2 @ D4 ) ) ) )
& ( ord_less_eq @ A @ C2 @ D4 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4184_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_4185_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_4186_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) )
= ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_4187_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ N )
= ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_4188_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_4189_sum__shift__lb__Suc0__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,K: nat] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_4190_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_4191_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_4192_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_4193_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ M )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_4194_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I4 ) ) @ ( F2 @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_4195_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_4196_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( F2 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_4197_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I4 ) @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( A2 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_4198_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,A2: nat,B2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( plus_plus @ A @ ( F2 @ A5 ) )
@ A2
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_4199_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A,P5: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P5 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_4200_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_4201_sum__up__index__split,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ M ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% sum_up_index_split
thf(fact_4202_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_4203_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_4204_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_4205_summable__partial__sum__bound,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,E2: real] :
( ( summable @ A @ F2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ~ ! [N8: nat] :
~ ! [M2: nat] :
( ( ord_less_eq @ nat @ N8 @ M2 )
=> ! [N9: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N9 ) ) ) @ E2 ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_4206_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_4207_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_4208_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,K: A] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= K ) )
= ( ( ( C2 @ ( zero_zero @ nat ) )
= K )
& ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) )
=> ( ( C2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_4209_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_4210_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_4211_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_4212_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,D4: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D4 ) ) ) ) ) ).
% double_arith_series
thf(fact_4213_arith__series__nat,axiom,
! [A2: nat,D4: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ A2 @ ( times_times @ nat @ I4 @ D4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ nat @ N @ D4 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_4214_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_4215_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_4216_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,D4: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D4 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_4217_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_4218_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ 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 @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_4219_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ Y @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ ( minus_minus @ nat @ I4 @ J3 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_4220_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_4221_VEBT__internal_Oinrange,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 ) ) ) ) ) ).
% VEBT_internal.inrange
thf(fact_4222_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ M @ N ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M @ ( minus_minus @ int @ M @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_4223_push__bit__numeral__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_4224_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_4225_push__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% push_bit_negative_int_iff
thf(fact_4226_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_4227_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se4730199178511100633sh_bit @ A @ N @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_4228_push__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A2 ) ) ) ).
% push_bit_push_bit
thf(fact_4229_Ints__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ B )
& ( ring_1 @ B ) )
=> ! [A3: set @ A,F2: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ B @ ( F2 @ X3 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_prod
thf(fact_4230_push__bit__and,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_and
thf(fact_4231_push__bit__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_xor
thf(fact_4232_concat__bit__of__zero__1,axiom,
! [N: nat,L: int] :
( ( bit_concat_bit @ N @ ( zero_zero @ int ) @ L )
= ( bit_se4730199178511100633sh_bit @ int @ N @ L ) ) ).
% concat_bit_of_zero_1
thf(fact_4233_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [Uu: B] : ( one_one @ A )
@ A3 )
= ( one_one @ A ) ) ) ).
% prod.neutral_const
thf(fact_4234_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F2 @ X2 ) )
@ A3 ) ) ) ).
% of_nat_prod
thf(fact_4235_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F2: B > int,A3: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( ring_1_of_int @ A @ ( F2 @ X2 ) )
@ A3 ) ) ) ).
% of_int_prod
thf(fact_4236_of__real__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2191834092415804123ebra_1 @ A ) )
=> ! [F2: B > real,S: set @ B] :
( ( real_Vector_of_real @ A @ ( groups7121269368397514597t_prod @ B @ real @ F2 @ S ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( real_Vector_of_real @ A @ ( F2 @ X2 ) )
@ S ) ) ) ).
% of_real_prod
thf(fact_4237_push__bit__Suc__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ K ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_4238_push__bit__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_4239_push__bit__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ K ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_numeral
thf(fact_4240_push__bit__minus__one__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_4241_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_4242_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A2 )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_4243_push__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_of_1
thf(fact_4244_even__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) )
= ( ( N
!= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_push_bit_iff
thf(fact_4245_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_4246_push__bit__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_4247_prod_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [H2: B > C,S2: set @ B,T5: set @ C,G: C > A] :
( ( bij_betw @ B @ C @ H2 @ S2 @ T5 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G @ ( H2 @ X2 ) )
@ S2 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T5 ) ) ) ) ).
% prod.reindex_bij_betw
thf(fact_4248_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H2: B > A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ A3 ) ) ) ) ).
% prod.distrib
thf(fact_4249_prod__dvd__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ! [A6: B] :
( ( member @ B @ A6 @ A3 )
=> ( dvd_dvd @ A @ ( F2 @ A6 ) @ ( G @ A6 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod_dvd_prod
thf(fact_4250_prod_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > C > A,B3: set @ C,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [I4: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ I4 ) @ B3 )
@ A3 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [J3: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [I4: B] : ( G @ I4 @ J3 )
@ A3 )
@ B3 ) ) ) ).
% prod.swap
thf(fact_4251_of__nat__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ M @ N ) )
= ( bit_se4730199178511100633sh_bit @ A @ M @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_push_bit
thf(fact_4252_push__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ N @ M ) ) ) ) ).
% push_bit_of_nat
thf(fact_4253_prod__power__distrib,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ B )
=> ! [F2: A > B,A3: set @ A,N: nat] :
( ( power_power @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ N )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ A3 ) ) ) ).
% prod_power_distrib
thf(fact_4254_abs__prod,axiom,
! [A: $tType,B: $tType] :
( ( linordered_field @ A )
=> ! [F2: B > A,A3: set @ B] :
( ( abs_abs @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( abs_abs @ A @ ( F2 @ X2 ) )
@ A3 ) ) ) ).
% abs_prod
thf(fact_4255_prod__dividef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F2: B > A,G: B > A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A3 )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod_dividef
thf(fact_4256_mod__prod__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F2: B > A,A2: A,A3: set @ B] :
( ( modulo_modulo @ A
@ ( groups7121269368397514597t_prod @ B @ A
@ ^ [I4: B] : ( modulo_modulo @ A @ ( F2 @ I4 ) @ A2 )
@ A3 )
@ A2 )
= ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ A2 ) ) ) ).
% mod_prod_eq
thf(fact_4257_push__bit__minus,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) ) ) ) ).
% push_bit_minus
thf(fact_4258_push__bit__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_add
thf(fact_4259_prod__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V8999393235501362500lgebra @ B )
& ( comm_semiring_1 @ B ) )
=> ! [F2: A > B,A3: set @ A] :
( ( groups7121269368397514597t_prod @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ A3 )
= ( real_V7770717601297561774m_norm @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) ) ) ) ).
% prod_norm
thf(fact_4260_push__bit__nat__eq,axiom,
! [N: nat,K: int] :
( ( bit_se4730199178511100633sh_bit @ nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) ) ) ).
% push_bit_nat_eq
thf(fact_4261_push__bit__of__int,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,K: int] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( ring_1_of_int @ A @ K ) )
= ( ring_1_of_int @ A @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) ) ) ) ).
% push_bit_of_int
thf(fact_4262_norm__prod__le,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [F2: B > A,A3: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) )
@ ( groups7121269368397514597t_prod @ B @ real
@ ^ [A5: B] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ A5 ) )
@ A3 ) ) ) ).
% norm_prod_le
thf(fact_4263_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ).
% prod_nonneg
thf(fact_4264_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) )
& ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod_mono
thf(fact_4265_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ).
% prod_pos
thf(fact_4266_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ).
% prod_ge_1
thf(fact_4267_push__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N ) @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) ) ) ) ).
% push_bit_take_bit
thf(fact_4268_take__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ M @ N ) @ A2 ) ) ) ) ).
% take_bit_push_bit
thf(fact_4269_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_4270_power__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: A,F2: B > nat,A3: set @ B] :
( ( power_power @ A @ C2 @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [A5: B] : ( power_power @ A @ C2 @ ( F2 @ A5 ) )
@ A3 ) ) ) ).
% power_sum
thf(fact_4271_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_4272_flip__bit__nat__def,axiom,
( ( bit_se8732182000553998342ip_bit @ nat )
= ( ^ [M3: nat,N3: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N3 @ ( bit_se4730199178511100633sh_bit @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ).
% flip_bit_nat_def
thf(fact_4273_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) )
& ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_4274_aset_I2_J,axiom,
! [D: int,A3: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D ) ) ) )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
=> ( ( P @ ( plus_plus @ int @ X6 @ D ) )
| ( Q @ ( plus_plus @ int @ X6 @ D ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_4275_aset_I1_J,axiom,
! [D: int,A3: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D ) ) ) )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
=> ( ( P @ ( plus_plus @ int @ X6 @ D ) )
& ( Q @ ( plus_plus @ int @ X6 @ D ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_4276_bset_I2_J,axiom,
! [D: int,B3: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D ) ) ) )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
=> ( ( P @ ( minus_minus @ int @ X6 @ D ) )
| ( Q @ ( minus_minus @ int @ X6 @ D ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_4277_bset_I1_J,axiom,
! [D: int,B3: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D ) ) ) )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
=> ( ( P @ ( minus_minus @ int @ X6 @ D ) )
& ( Q @ ( minus_minus @ int @ X6 @ D ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_4278_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_4279_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_4280_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_4281_bit__push__bit__iff__nat,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q3 ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ Q3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_4282_concat__bit__eq,axiom,
( bit_concat_bit
= ( ^ [N3: nat,K3: int,L2: int] : ( plus_plus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K3 ) @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ L2 ) ) ) ) ).
% concat_bit_eq
thf(fact_4283_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N3: nat,A5: A] : ( bit_se5824344971392196577ns_xor @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_4284_flip__bit__int__def,axiom,
( ( bit_se8732182000553998342ip_bit @ int )
= ( ^ [N3: nat,K3: int] : ( bit_se5824344971392196577ns_xor @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ).
% flip_bit_int_def
thf(fact_4285_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_4286_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_4287_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_4288_aset_I10_J,axiom,
! [D4: int,D: int,A3: set @ int,T2: int] :
( ( dvd_dvd @ int @ D4 @ D )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D4 @ ( plus_plus @ int @ X6 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D4 @ ( plus_plus @ int @ ( plus_plus @ int @ X6 @ D ) @ T2 ) ) ) ) ) ).
% aset(10)
thf(fact_4289_aset_I9_J,axiom,
! [D4: int,D: int,A3: set @ int,T2: int] :
( ( dvd_dvd @ int @ D4 @ D )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D4 @ ( plus_plus @ int @ X6 @ T2 ) )
=> ( dvd_dvd @ int @ D4 @ ( plus_plus @ int @ ( plus_plus @ int @ X6 @ D ) @ T2 ) ) ) ) ) ).
% aset(9)
thf(fact_4290_bset_I10_J,axiom,
! [D4: int,D: int,B3: set @ int,T2: int] :
( ( dvd_dvd @ int @ D4 @ D )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D4 @ ( plus_plus @ int @ X6 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D4 @ ( plus_plus @ int @ ( minus_minus @ int @ X6 @ D ) @ T2 ) ) ) ) ) ).
% bset(10)
thf(fact_4291_bset_I9_J,axiom,
! [D4: int,D: int,B3: set @ int,T2: int] :
( ( dvd_dvd @ int @ D4 @ D )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D4 @ ( plus_plus @ int @ X6 @ T2 ) )
=> ( dvd_dvd @ int @ D4 @ ( plus_plus @ int @ ( minus_minus @ int @ X6 @ D ) @ T2 ) ) ) ) ) ).
% bset(9)
thf(fact_4292_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_4293_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_4294_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ M )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_4295_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_4296_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
=> ( ( set_or1337092689740270186AtMost @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
= ( insert @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ ( set_or1337092689740270186AtMost @ int @ M @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_4297_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_4298_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ) ).
% fact_prod
thf(fact_4299_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I4 ) @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( A2 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_4300_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_4301_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F2: nat > A,A2: nat,B2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( times_times @ A @ ( F2 @ A5 ) )
@ A2
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_4302_push__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N @ M ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) ) ) ) ) ).
% push_bit_mask_eq
thf(fact_4303_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N3: nat,A5: A] : ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_4304_unset__bit__int__def,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N3: nat,K3: int] : ( bit_se5824344872417868541ns_and @ int @ K3 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_4305_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A,P5: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P5 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_4306_periodic__finite__ex,axiom,
! [D4: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D4 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P @ X2 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_4307_aset_I7_J,axiom,
! [D: int,A3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X6 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X6 @ D ) ) ) ) ) ).
% aset(7)
thf(fact_4308_aset_I5_J,axiom,
! [D: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ T2 @ A3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X6 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X6 @ D ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_4309_aset_I4_J,axiom,
! [D: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ T2 @ A3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X6 != T2 )
=> ( ( plus_plus @ int @ X6 @ D )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_4310_aset_I3_J,axiom,
! [D: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X6 = T2 )
=> ( ( plus_plus @ int @ X6 @ D )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_4311_bset_I7_J,axiom,
! [D: int,T2: int,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ T2 @ B3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X6 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X6 @ D ) ) ) ) ) ) ).
% bset(7)
thf(fact_4312_bset_I5_J,axiom,
! [D: int,B3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X6 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X6 @ D ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_4313_bset_I4_J,axiom,
! [D: int,T2: int,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ T2 @ B3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X6 != T2 )
=> ( ( minus_minus @ int @ X6 @ D )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_4314_bset_I3_J,axiom,
! [D: int,T2: int,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X6 = T2 )
=> ( ( minus_minus @ int @ X6 @ D )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_4315_push__bit__nat__def,axiom,
( ( bit_se4730199178511100633sh_bit @ nat )
= ( ^ [N3: nat,M3: nat] : ( times_times @ nat @ M3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% push_bit_nat_def
thf(fact_4316_push__bit__int__def,axiom,
( ( bit_se4730199178511100633sh_bit @ int )
= ( ^ [N3: nat,K3: int] : ( times_times @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% push_bit_int_def
thf(fact_4317_norm__prod__diff,axiom,
! [A: $tType,I5: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [I6: set @ I5,Z3: I5 > A,W2: I5 > A] :
( ! [I2: I5] :
( ( member @ I5 @ I2 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( Z3 @ I2 ) ) @ ( one_one @ real ) ) )
=> ( ! [I2: I5] :
( ( member @ I5 @ I2 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( W2 @ I2 ) ) @ ( one_one @ real ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( groups7121269368397514597t_prod @ I5 @ A @ Z3 @ I6 ) @ ( groups7121269368397514597t_prod @ I5 @ A @ W2 @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ I5 @ real
@ ^ [I4: I5] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( Z3 @ I4 ) @ ( W2 @ I4 ) ) )
@ I6 ) ) ) ) ) ).
% norm_prod_diff
thf(fact_4318_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_4319_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N3: nat,A5: A] : ( times_times @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_4320_exp__dvdE,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A2 )
=> ~ ! [B6: A] :
( A2
!= ( bit_se4730199178511100633sh_bit @ A @ N @ B6 ) ) ) ) ).
% exp_dvdE
thf(fact_4321_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( semiring_char_0_fact @ nat @ M )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_4322_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I4: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I4 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I4 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_4323_aset_I8_J,axiom,
! [D: int,A3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X6 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X6 @ D ) ) ) ) ) ).
% aset(8)
thf(fact_4324_aset_I6_J,axiom,
! [D: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X6
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X6 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X6 @ D ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_4325_bset_I8_J,axiom,
! [D: int,T2: int,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B3 )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X6 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X6 @ D ) ) ) ) ) ) ).
% bset(8)
thf(fact_4326_bset_I6_J,axiom,
! [D: int,B3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ! [X6: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B3 )
=> ( X6
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X6 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X6 @ D ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_4327_cpmi,axiom,
! [D: int,P: int > $o,P4: int > $o,B3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B3 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
& ( P4 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
& ? [Y4: int] :
( ( member @ int @ Y4 @ B3 )
& ( P @ ( plus_plus @ int @ Y4 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_4328_cppi,axiom,
! [D: int,P: int > $o,P4: int > $o,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less @ int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P4 @ X3 )
= ( P4 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
& ( P4 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D ) )
& ? [Y4: int] :
( ( member @ int @ Y4 @ A3 )
& ( P @ ( minus_minus @ int @ Y4 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_4329_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_4330_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_4331_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N3 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_4332_fact__div__fact,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_4333_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_4334_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_4335_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_4336_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P5: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
=> ( ( ord_less_eq @ nat @ K @ P5 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P5 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_4337_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A5: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A5 ) @ N3 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A5 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N3 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A5 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_4338_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ A2 @ ( suc @ K ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_4339_bij__betw__nth__root__unity,axiom,
! [C2: complex,N: nat] :
( ( C2
!= ( 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 @ C2 ) ) ) @ ( cis @ ( divide_divide @ real @ ( arg @ C2 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) )
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= ( one_one @ complex ) ) )
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= C2 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_4340_VEBT__internal_Oinsert__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 ) ) ) ) ) ).
% VEBT_internal.insert_corr
thf(fact_4341_set__encode__def,axiom,
( nat_set_encode
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% set_encode_def
thf(fact_4342_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X8: nat > real] :
! [J3: nat] :
? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M9 @ N3 )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_4343_UnCI,axiom,
! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ A3 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_4344_Un__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C2 @ A3 )
| ( member @ A @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_4345_Un__empty,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_4346_Un__subset__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_4347_Un__insert__right,axiom,
! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_4348_Un__insert__left,axiom,
! [A: $tType,A2: A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C3 )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_4349_Un__Diff__cancel2,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) @ A3 )
= ( sup_sup @ ( set @ A ) @ B3 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_4350_Un__Diff__cancel,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_Diff_cancel
thf(fact_4351_real__root__zero,axiom,
! [N: nat] :
( ( root @ N @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_root_zero
thf(fact_4352_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_4353_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_4354_root__0,axiom,
! [X: real] :
( ( root @ ( zero_zero @ nat ) @ X )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_4355_Compl__Diff__eq,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ B3 ) ) ).
% Compl_Diff_eq
thf(fact_4356_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_4357_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_4358_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_4359_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_4360_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_4361_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_4362_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_4363_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_4364_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_4365_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_4366_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_4367_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_4368_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_4369_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_4370_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_4371_int__prod,axiom,
! [B: $tType,F2: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F2 @ X2 ) )
@ A3 ) ) ).
% int_prod
thf(fact_4372_Un__Diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ C3 ) @ ( minus_minus @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_Diff
thf(fact_4373_insert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A5: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] : X2 = A5 ) ) ) ) ).
% insert_def
thf(fact_4374_real__root__inverse,axiom,
! [N: nat,X: real] :
( ( root @ N @ ( inverse_inverse @ real @ X ) )
= ( inverse_inverse @ real @ ( root @ N @ X ) ) ) ).
% real_root_inverse
thf(fact_4375_Un__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Un_empty_right
thf(fact_4376_Un__empty__left,axiom,
! [A: $tType,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_4377_Un__mono,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,B3: set @ A,D: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ D )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% Un_mono
thf(fact_4378_Un__least,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 ) ) ) ).
% Un_least
thf(fact_4379_Un__upper1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_upper1
thf(fact_4380_Un__upper2,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_upper2
thf(fact_4381_Un__absorb1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_4382_Un__absorb2,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= A3 ) ) ).
% Un_absorb2
thf(fact_4383_subset__UnE,axiom,
! [A: $tType,C3: set @ A,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ A3 )
=> ! [B10: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B10 @ B3 )
=> ( C3
!= ( sup_sup @ ( set @ A ) @ A8 @ B10 ) ) ) ) ) ).
% subset_UnE
thf(fact_4384_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_4385_real__root__mult__exp,axiom,
! [M: nat,N: nat,X: real] :
( ( root @ ( times_times @ nat @ M @ N ) @ X )
= ( root @ M @ ( root @ N @ X ) ) ) ).
% real_root_mult_exp
thf(fact_4386_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_4387_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
| ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_4388_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_4389_UnE,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% UnE
thf(fact_4390_UnI1,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_4391_UnI2,axiom,
! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
( ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_4392_bex__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B3 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_4393_ball__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B3 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_4394_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_assoc
thf(fact_4395_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_4396_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] : ( sup_sup @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_4397_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_left_absorb
thf(fact_4398_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_4399_real__root__commute,axiom,
! [M: nat,N: nat,X: real] :
( ( root @ M @ ( root @ N @ X ) )
= ( root @ N @ ( root @ M @ X ) ) ) ).
% real_root_commute
thf(fact_4400_real__root__divide,axiom,
! [N: nat,X: real,Y: real] :
( ( root @ N @ ( divide_divide @ real @ X @ Y ) )
= ( divide_divide @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).
% real_root_divide
thf(fact_4401_real__root__minus,axiom,
! [N: nat,X: real] :
( ( root @ N @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( root @ N @ X ) ) ) ).
% real_root_minus
thf(fact_4402_Collect__imp__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_4403_real__root__pos__pos__le,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_4404_ivl__disj__un__two__touch_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U2 ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_4405_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A5: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_4406_Un__singleton__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_4407_singleton__Un__iff,axiom,
! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_4408_Diff__subset__conv,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ C3 )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_4409_Diff__partition,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) )
= B3 ) ) ).
% Diff_partition
thf(fact_4410_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_4411_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_4412_real__root__power,axiom,
! [N: nat,X: real,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X @ K ) )
= ( power_power @ real @ ( root @ N @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_4413_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_4414_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_4415_prod__int__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_4416_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_4417_real__root__strict__decreasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_4418_sqrt__def,axiom,
( sqrt
= ( root @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% sqrt_def
thf(fact_4419_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_4420_ivl__disj__un__one_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U2 ) )
= ( set_ord_atMost @ A @ U2 ) ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_4421_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [U2: A] :
( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U2 ) @ ( insert @ A @ U2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_ord_atMost @ A @ U2 ) ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_4422_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_4423_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_4424_real__root__strict__increasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_4425_real__root__decreasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_4426_odd__real__root__pow,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ).
% odd_real_root_pow
thf(fact_4427_odd__real__root__unique,axiom,
! [N: nat,Y: real,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ( power_power @ real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ).
% odd_real_root_unique
thf(fact_4428_odd__real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X @ N ) )
= X ) ) ).
% odd_real_root_power_cancel
thf(fact_4429_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_4430_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_4431_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_4432_real__root__increasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_4433_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_4434_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_4435_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M9 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_4436_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M10 @ M4 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M10 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X5 @ M4 ) @ ( X5 @ N2 ) ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X5 ) ) ) ).
% CauchyI
thf(fact_4437_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A,E2: real] :
( ( topolo3814608138187158403Cauchy @ A @ X5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M8: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M8 @ M2 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ M8 @ N9 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X5 @ M2 ) @ ( X5 @ N9 ) ) ) @ E2 ) ) ) ) ) ) ).
% CauchyD
thf(fact_4438_log__root,axiom,
! [N: nat,A2: real,B2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ B2 @ ( root @ N @ A2 ) )
= ( divide_divide @ real @ ( log @ B2 @ A2 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_root
thf(fact_4439_log__base__root,axiom,
! [N: nat,B2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( log @ ( root @ N @ B2 ) @ X )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_4440_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_4441_split__root,axiom,
! [P: real > $o,N: nat,X: real] :
( ( P @ ( root @ N @ X ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ! [Y4: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y4 ) @ ( power_power @ real @ ( abs_abs @ real @ Y4 ) @ N ) )
= X )
=> ( P @ Y4 ) ) ) ) ) ).
% split_root
thf(fact_4442_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_4443_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_4444_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_4445_set__vebt__insert,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ n ) )
=> ( ( vEBT_set_vebt @ ( vEBT_vebt_insert @ T2 @ X ) )
= ( sup_sup @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% set_vebt_insert
thf(fact_4446_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B2: A,A3: set @ A,F2: A > B,A9: set @ B] :
( ~ ( member @ A @ B2 @ A3 )
=> ( ~ ( member @ B @ ( F2 @ B2 ) @ A9 )
=> ( ( bij_betw @ A @ B @ F2 @ A3 @ A9 )
= ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A9 @ ( insert @ B @ ( F2 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_4447_set__vebt__buildup,axiom,
! [I: nat] :
( ( vEBT_set_vebt @ ( vEBT_vebt_buildup @ I ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_vebt_buildup
thf(fact_4448_set__vebt__equal,axiom,
! [T_1: vEBT_VEBT,T_2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T_1 @ n )
=> ( ( vEBT_invar_vebt @ T_2 @ n )
=> ( ( T_1 = T_2 )
= ( ( vEBT_set_vebt @ T_1 )
= ( vEBT_set_vebt @ T_2 ) ) ) ) ) ).
% set_vebt_equal
thf(fact_4449_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ ^ [X2: A] : ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_4450_sup__Un__eq,axiom,
! [A: $tType,R2: set @ A,S2: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R2 )
@ ^ [X2: A] : ( member @ A @ X2 @ S2 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ R2 @ S2 ) ) ) ) ).
% sup_Un_eq
thf(fact_4451_vebt__inst_Oset__vebt__equal,axiom,
! [T_1: vEBT_VEBT,N: nat,T_2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T_1 @ N )
=> ( ( vEBT_invar_vebt @ T_2 @ N )
=> ( ( T_1 = T_2 )
= ( ( vEBT_set_vebt @ T_1 )
= ( vEBT_set_vebt @ T_2 ) ) ) ) ) ).
% vebt_inst.set_vebt_equal
thf(fact_4452_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_4453_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_4454_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A2 @ X )
=> ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% le_supE
thf(fact_4455_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X )
=> ( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X ) ) ) ) ).
% le_supI
thf(fact_4456_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_4457_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_4458_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_4459_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_4460_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,D4: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D4 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D4 ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_4461_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,C2: A,B2: A,D4: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D4 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C2 @ D4 ) ) ) ) ) ).
% sup_mono
thf(fact_4462_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_4463_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y4: A] :
( ( sup_sup @ A @ X2 @ Y4 )
= Y4 ) ) ) ) ).
% le_iff_sup
thf(fact_4464_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_4465_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% sup.orderI
thf(fact_4466_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F2: A > A > A,X: A,Y: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_4467_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb1
thf(fact_4468_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_4469_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_4470_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_4471_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_4472_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_4473_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% sup.order_iff
thf(fact_4474_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_4475_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_4476_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_4477_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_4478_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_4479_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_4480_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_4481_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ A2 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_4482_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( A5
= ( sup_sup @ A @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_4483_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_4484_sup_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb4
thf(fact_4485_sup_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb3
thf(fact_4486_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A2: A] :
( ( ord_less @ A @ X @ B2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI2
thf(fact_4487_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less @ A @ X @ A2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI1
thf(fact_4488_vebt__inst_Oset__vebt__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_set_vebt @ ( vEBT_vebt_insert @ T2 @ X ) )
= ( sup_sup @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% vebt_inst.set_vebt_insert
thf(fact_4489_VEBT__internal_Oinsert__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 ) ) ) ) ) ).
% VEBT_internal.insert_correct
thf(fact_4490_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B2: A,A3: set @ A,F2: A > B,A9: set @ B] :
( ~ ( member @ A @ B2 @ A3 )
=> ( ~ ( member @ B @ ( F2 @ B2 ) @ A9 )
=> ( ( bij_betw @ A @ B @ F2 @ A3 @ A9 )
=> ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A9 @ ( insert @ B @ ( F2 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_4491_set__vebt__minNull,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( vEBT_VEBT_minNull @ T2 )
= ( ( vEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% set_vebt_minNull
thf(fact_4492_set__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( union @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_union
thf(fact_4493_VEBT__internal_Oinsert_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 ) ) ) ) ) ) ).
% VEBT_internal.insert'_correct
thf(fact_4494_set__vebt__delete,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( vEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% set_vebt_delete
thf(fact_4495_invar__vebt__delete,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T2 @ X ) @ n ) ) ).
% invar_vebt_delete
thf(fact_4496_IntI,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_4497_Int__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C2 @ A3 )
& ( member @ A @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_4498_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_4499_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% inf.bounded_iff
thf(fact_4500_Int__subset__iff,axiom,
! [A: $tType,C3: set @ A,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C3 @ A3 )
& ( ord_less_eq @ ( set @ A ) @ C3 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_4501_Int__insert__right__if1,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_4502_Int__insert__right__if0,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_4503_insert__inter__insert,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_4504_Int__insert__left__if1,axiom,
! [A: $tType,A2: A,C3: set @ A,B3: set @ A] :
( ( member @ A @ A2 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C3 )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_4505_Int__insert__left__if0,axiom,
! [A: $tType,A2: A,C3: set @ A,B3: set @ A] :
( ~ ( member @ A @ A2 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_4506_Int__Un__eq_I4_J,axiom,
! [A: $tType,T5: set @ A,S2: set @ A] :
( ( sup_sup @ ( set @ A ) @ T5 @ ( inf_inf @ ( set @ A ) @ S2 @ T5 ) )
= T5 ) ).
% Int_Un_eq(4)
thf(fact_4507_Int__Un__eq_I3_J,axiom,
! [A: $tType,S2: set @ A,T5: set @ A] :
( ( sup_sup @ ( set @ A ) @ S2 @ ( inf_inf @ ( set @ A ) @ S2 @ T5 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_4508_Int__Un__eq_I2_J,axiom,
! [A: $tType,S2: set @ A,T5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S2 @ T5 ) @ T5 )
= T5 ) ).
% Int_Un_eq(2)
thf(fact_4509_Int__Un__eq_I1_J,axiom,
! [A: $tType,S2: set @ A,T5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S2 @ T5 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_4510_Un__Int__eq_I4_J,axiom,
! [A: $tType,T5: set @ A,S2: set @ A] :
( ( inf_inf @ ( set @ A ) @ T5 @ ( sup_sup @ ( set @ A ) @ S2 @ T5 ) )
= T5 ) ).
% Un_Int_eq(4)
thf(fact_4511_Un__Int__eq_I3_J,axiom,
! [A: $tType,S2: set @ A,T5: set @ A] :
( ( inf_inf @ ( set @ A ) @ S2 @ ( sup_sup @ ( set @ A ) @ S2 @ T5 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_4512_Un__Int__eq_I2_J,axiom,
! [A: $tType,S2: set @ A,T5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S2 @ T5 ) @ T5 )
= T5 ) ).
% Un_Int_eq(2)
thf(fact_4513_Un__Int__eq_I1_J,axiom,
! [A: $tType,S2: set @ A,T5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S2 @ T5 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_4514_disjoint__insert_I2_J,axiom,
! [A: $tType,A3: set @ A,B2: A,B3: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B3 ) ) )
= ( ~ ( member @ A @ B2 @ A3 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_4515_disjoint__insert_I1_J,axiom,
! [A: $tType,B3: set @ A,A2: A,A3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A2 @ B3 )
& ( ( inf_inf @ ( set @ A ) @ B3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_4516_insert__disjoint_I2_J,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ B3 ) )
= ( ~ ( member @ A @ A2 @ B3 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_4517_insert__disjoint_I1_J,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A2 @ B3 )
& ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_4518_Diff__disjoint,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_4519_Compl__disjoint2,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_4520_Compl__disjoint,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_4521_Diff__Compl,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ).
% Diff_Compl
thf(fact_4522_Un__Int__crazy,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) @ ( inf_inf @ ( set @ A ) @ C3 @ A3 ) )
= ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) @ ( sup_sup @ ( set @ A ) @ C3 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_4523_Int__Un__distrib,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_4524_Un__Int__distrib,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ ( sup_sup @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_4525_Int__Un__distrib2,axiom,
! [A: $tType,B3: set @ A,C3: set @ A,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) @ A3 )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B3 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C3 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_4526_Un__Int__distrib2,axiom,
! [A: $tType,B3: set @ A,C3: set @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) @ A3 )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B3 @ A3 ) @ ( sup_sup @ ( set @ A ) @ C3 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_4527_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_4528_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ A2 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_4529_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( A5
= ( inf_inf @ A @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_4530_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_4531_inf_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb4
thf(fact_4532_inf_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb3
thf(fact_4533_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A2: A] :
( ( ord_less @ A @ B2 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% less_infI2
thf(fact_4534_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less @ A @ A2 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% less_infI1
thf(fact_4535_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_4536_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_4537_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_4538_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_4539_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A2 )
=> ~ ( ord_less_eq @ A @ X @ B2 ) ) ) ) ).
% le_infE
thf(fact_4540_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_4541_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A,D4: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D4 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C2 @ D4 ) ) ) ) ) ).
% inf_mono
thf(fact_4542_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% le_infI1
thf(fact_4543_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% le_infI2
thf(fact_4544_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2
= ( inf_inf @ A @ A2 @ B2 ) ) ) ) ).
% inf.orderE
thf(fact_4545_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( inf_inf @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% inf.orderI
thf(fact_4546_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F2: A > A > A,X: A,Y: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( F2 @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( F2 @ 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 @ ( F2 @ Y3 @ Z ) ) ) )
=> ( ( inf_inf @ A @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_4547_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y4: A] :
( ( inf_inf @ A @ X2 @ Y4 )
= X2 ) ) ) ) ).
% le_iff_inf
thf(fact_4548_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb1
thf(fact_4549_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb2
thf(fact_4550_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_4551_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_4552_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% inf.boundedE
thf(fact_4553_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ) ).
% inf.boundedI
thf(fact_4554_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_4555_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( A5
= ( inf_inf @ A @ A5 @ B5 ) ) ) ) ) ).
% inf.order_iff
thf(fact_4556_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ A2 ) ) ).
% inf.cobounded1
thf(fact_4557_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 ) ) ).
% inf.cobounded2
thf(fact_4558_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( inf_inf @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_4559_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( inf_inf @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_4560_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI1
thf(fact_4561_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI2
thf(fact_4562_Diff__Int__distrib2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ C3 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_4563_Diff__Int__distrib,axiom,
! [A: $tType,C3: set @ A,A3: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C3 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C3 @ B3 ) ) ) ).
% Diff_Int_distrib
thf(fact_4564_Diff__Diff__Int,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ).
% Diff_Diff_Int
thf(fact_4565_Diff__Int2,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) @ B3 ) ) ).
% Diff_Int2
thf(fact_4566_Int__Diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Int_Diff
thf(fact_4567_disjoint__iff__not__equal,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ B3 )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_4568_Int__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_4569_Int__empty__left,axiom,
! [A: $tType,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_4570_disjoint__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ~ ( member @ A @ X2 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_4571_Int__emptyI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ~ ( member @ A @ X3 @ B3 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_4572_Int__mono,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,B3: set @ A,D: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ D )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( inf_inf @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_4573_Int__lower1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ).
% Int_lower1
thf(fact_4574_Int__lower2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_4575_Int__absorb1,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_4576_Int__absorb2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= A3 ) ) ).
% Int_absorb2
thf(fact_4577_Int__greatest,axiom,
! [A: $tType,C3: set @ A,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ C3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_4578_Int__Collect__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B3 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_4579_Int__insert__left,axiom,
! [A: $tType,A2: A,C3: set @ A,B3: set @ A] :
( ( ( member @ A @ A2 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C3 )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) )
& ( ~ ( member @ A @ A2 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_4580_Int__insert__right,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: set @ A] :
( ( ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) )
& ( ~ ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_4581_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_4582_Int__Collect,axiom,
! [A: $tType,X: A,A3: set @ A,P: A > $o] :
( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X @ A3 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_4583_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_4584_IntE,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ( ( member @ A @ C2 @ A3 )
=> ~ ( member @ A @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_4585_IntD1,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
=> ( member @ A @ C2 @ A3 ) ) ).
% IntD1
thf(fact_4586_IntD2,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
=> ( member @ A @ C2 @ B3 ) ) ).
% IntD2
thf(fact_4587_Int__assoc,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Int_assoc
thf(fact_4588_Int__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_4589_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] : ( inf_inf @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_4590_Int__left__absorb,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ).
% Int_left_absorb
thf(fact_4591_Int__left__commute,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_4592_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_4593_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_4594_VEBT__internal_OminNull__delete__time__bound_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X ) )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X ) @ ( one_one @ nat ) ) ) ) ).
% VEBT_internal.minNull_delete_time_bound'
thf(fact_4595_Int__Diff__disjoint,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_4596_Diff__triv,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ B3 )
= A3 ) ) ).
% Diff_triv
thf(fact_4597_Un__Int__assoc__eq,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) )
= ( ord_less_eq @ ( set @ A ) @ C3 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_4598_Diff__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Diff_Un
thf(fact_4599_Diff__Int,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Diff_Int
thf(fact_4600_Int__Diff__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_4601_Un__Diff__Int,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_4602_Compl__Int,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ).
% Compl_Int
thf(fact_4603_Compl__Un,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ).
% Compl_Un
thf(fact_4604_Diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] : ( inf_inf @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B4 ) ) ) ) ).
% Diff_eq
thf(fact_4605_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_4606_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P5: A,Q3: A,R3: A] :
( ( ord_less_eq @ A @ P5 @ ( sup_sup @ A @ Q3 @ R3 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P5 @ ( uminus_uminus @ A @ Q3 ) ) @ R3 ) ) ) ).
% sup_neg_inf
thf(fact_4607_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_4608_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_4609_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_4610_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,C3: set @ B,G: A > B,B3: set @ A,D: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A3 @ C3 )
=> ( ( bij_betw @ A @ B @ G @ B3 @ D )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C3 @ D )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ A3 ) @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup @ ( set @ A ) @ A3 @ B3 )
@ ( sup_sup @ ( set @ B ) @ C3 @ D ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_4611_VEBT__internal_OminNull__delete__time__bound,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X ) )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% VEBT_internal.minNull_delete_time_bound
thf(fact_4612_vebt__inst_Oset__vebt__minNull,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ T2 )
= ( ( vEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% vebt_inst.set_vebt_minNull
thf(fact_4613_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,K: A] :
( ( ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_4614_vebt__inst_Oset__vebt__delete,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X ) )
= ( minus_minus @ ( set @ nat ) @ ( vEBT_set_vebt @ T2 ) @ ( insert @ nat @ X @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% vebt_inst.set_vebt_delete
thf(fact_4615_insertsimp,axiom,
! [T2: vEBT_VEBT,N: nat,L: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ T2 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ L ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ).
% insertsimp
thf(fact_4616_VEBT__internal_Odelete__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 ) ) ) ) ) ) ).
% VEBT_internal.delete_correct
thf(fact_4617_VEBT__internal_Odelete__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 ) ) ) ) ) ) ).
% VEBT_internal.delete_correct'
thf(fact_4618_set__encode__insert,axiom,
! [A3: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ~ ( member @ nat @ N @ A3 )
=> ( ( nat_set_encode @ ( insert @ nat @ N @ A3 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).
% set_encode_insert
thf(fact_4619_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_4620_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_4621_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F4: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ F4 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ F4 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ F4 )
=> ( ( F2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_4622_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% sum.infinite
thf(fact_4623_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U2: A] :
( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U2 ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less @ A @ I @ U2 ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_4624_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Icc_iff
thf(fact_4625_prod__zero__iff,axiom,
! [A: $tType,B: $tType] :
( ( semidom @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 )
= ( zero_zero @ A ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ( F2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_4626_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_4627_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,J: A,M: A,N: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
= ( ( ord_less_eq @ A @ J @ I )
| ( ( ord_less_eq @ A @ M @ I )
& ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% ivl_subset
thf(fact_4628_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_4629_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) )
= ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_4630_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ico_iff
thf(fact_4631_dvd__prodI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: set @ B,A2: B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ A2 @ A3 )
=> ( dvd_dvd @ A @ ( F2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% dvd_prodI
thf(fact_4632_dvd__prod__eqI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: set @ B,A2: B,B2: A,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ A2 @ A3 )
=> ( ( B2
= ( F2 @ A2 ) )
=> ( dvd_dvd @ A @ B2 @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_4633_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,N: A,M: A] :
( ( ord_less_eq @ A @ I @ N )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M ) @ ( set_or7035219750837199246ssThan @ A @ I @ N ) )
= ( set_or7035219750837199246ssThan @ A @ N @ M ) ) ) ) ).
% ivl_diff
thf(fact_4634_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S2 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_4635_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S2 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_4636_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S2 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S2 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_4637_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S2 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S2 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_4638_summable__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite
thf(fact_4639_summable__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A3: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ A3 )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A3 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite_set
thf(fact_4640_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum.insert
thf(fact_4641_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod.insert
thf(fact_4642_prod__pos__nat__iff,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_4643_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_4644_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,P: B > $o,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) @ ( F2 @ X2 ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_4645_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,F2: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_4646_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_4647_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_4648_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: set @ nat,C2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A3 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_4649_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: set @ nat,C2: nat > A,D4: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D4 @ I4 ) )
@ A3 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D4 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D4 @ I4 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_4650_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ ^ [X2: A] : ( member @ A @ X2 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_4651_inf__Int__eq,axiom,
! [A: $tType,R2: set @ A,S2: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R2 )
@ ^ [X2: A] : ( member @ A @ X2 @ S2 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ R2 @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_4652_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_4653_finite__less__ub,axiom,
! [F2: nat > nat,U2: nat] :
( ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( F2 @ N2 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ ( F2 @ N3 ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_4654_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B3: set @ C,G: B > C > A,R2: B > C > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ C @ B3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G @ X2 )
@ ( collect @ C
@ ^ [Y4: C] :
( ( member @ C @ Y4 @ B3 )
& ( R2 @ X2 @ Y4 ) ) ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y4: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G @ X2 @ Y4 )
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( R2 @ X2 @ Y4 ) ) ) )
@ B3 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_4655_subset__eq__atLeast0__lessThan__finite,axiom,
! [N4: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N4 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_4656_finite__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [Xs3: list @ A] :
( ( set2 @ A @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_4657_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ico
thf(fact_4658_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D4 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D4 )
=> ( B2 = D4 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_4659_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D4 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D4 )
=> ( A2 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_4660_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D4 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D4 ) )
= ( ( A2 = C2 )
& ( B2 = D4 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_4661_bounded__nat__set__is__finite,axiom,
! [N4: set @ nat,N: nat] :
( ! [X3: nat] :
( ( member @ nat @ X3 @ N4 )
=> ( ord_less @ nat @ X3 @ N ) )
=> ( finite_finite2 @ nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_4662_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N7: set @ nat] :
? [M3: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N7 )
=> ( ord_less @ nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_4663_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N7: set @ nat] :
? [M3: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N7 )
=> ( ord_less_eq @ nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_4664_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B3: set @ C,G: B > C > A,R2: B > C > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ C @ B3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G @ X2 )
@ ( collect @ C
@ ^ [Y4: C] :
( ( member @ C @ Y4 @ B3 )
& ( R2 @ X2 @ Y4 ) ) ) )
@ A3 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y4: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G @ X2 @ Y4 )
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( R2 @ X2 @ Y4 ) ) ) )
@ B3 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_4665_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D4 ) )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D4 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_4666_ivl__disj__un__two_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U2 ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_4667_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_4668_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_4669_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_4670_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X5: set @ A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ X5 )
& ( ord_less @ A @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite2 @ A @ X5 ) ) ) ) ).
% infinite_growing
thf(fact_4671_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S2 )
& ~ ? [Xa2: A] :
( ( member @ A @ Xa2 @ S2 )
& ( ord_less @ A @ Xa2 @ X3 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4672_sum__mono__inv,axiom,
! [A: $tType,I5: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F2: I5 > A,I6: set @ I5,G: I5 > A,I: I5] :
( ( ( groups7311177749621191930dd_sum @ I5 @ A @ F2 @ I6 )
= ( groups7311177749621191930dd_sum @ I5 @ A @ G @ I6 ) )
=> ( ! [I2: I5] :
( ( member @ I5 @ I2 @ I6 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) )
=> ( ( member @ I5 @ I @ I6 )
=> ( ( finite_finite2 @ I5 @ I6 )
=> ( ( F2 @ I )
= ( G @ I ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_4673_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Icc
thf(fact_4674_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ? [X6: B] :
( ( member @ B @ X6 @ A3 )
& ( ( F2 @ X6 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_4675_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_4676_summable__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N4: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N4 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N4 )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_finite
thf(fact_4677_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_4678_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_4679_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_4680_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( X @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( Y @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( plus_plus @ A @ ( X @ I4 ) @ ( Y @ I4 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4681_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( X @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( Y @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( times_times @ A @ ( X @ I4 ) @ ( Y @ I4 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4682_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_4683_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( zero_zero @ A ) )
@ A3 ) ) ) ) ).
% sum.inter_filter
thf(fact_4684_set__encode__inf,axiom,
! [A3: set @ nat] :
( ~ ( finite_finite2 @ nat @ A3 )
=> ( ( nat_set_encode @ A3 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_4685_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( one_one @ A ) )
@ A3 ) ) ) ) ).
% prod.inter_filter
thf(fact_4686_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A2 @ X2 )
& ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_4687_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A2: B,C2: B,B2: B,D4: B,G: B > A,H2: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D4 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D4 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D4 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_4688_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A2: B,C2: B,B2: B,D4: B,G: B > A,H2: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D4 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D4 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D4 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_4689_ivl__disj__un__two_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U2 ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_4690_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,P5: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P5 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_4691_ivl__disj__un__one_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U2 ) )
= ( set_ord_lessThan @ A @ U2 ) ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_4692_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,P5: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P5 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_4693_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,P5: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P5 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_4694_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,T2: set @ C,G: C > A,I: C > B,F2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T2 )
& ( ( I @ Xa2 )
= X3 )
& ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_4695_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( F2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_4696_sum__strict__mono__ex1,axiom,
! [A: $tType,I5: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: set @ I5,F2: I5 > A,G: I5 > A] :
( ( finite_finite2 @ I5 @ A3 )
=> ( ! [X3: I5] :
( ( member @ I5 @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X6: I5] :
( ( member @ I5 @ X6 @ A3 )
& ( ord_less @ A @ ( F2 @ X6 ) @ ( G @ X6 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I5 @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ I5 @ A @ G @ A3 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_4697_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R2: A > A > $o,S2: set @ B,H2: B > A,G: B > A] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X1: A,Y1: A,X23: A,Y23: A] :
( ( ( R2 @ X1 @ X23 )
& ( R2 @ Y1 @ Y23 ) )
=> ( R2 @ ( plus_plus @ A @ X1 @ Y1 ) @ ( plus_plus @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R2 @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_4698_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S2: set @ B,P: ( set @ B ) > $o,F2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S4: set @ B] :
( ( finite_finite2 @ B @ S4 )
=> ( ! [Y2: B] :
( ( member @ B @ Y2 @ S4 )
=> ( ord_less_eq @ A @ ( F2 @ Y2 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert @ B @ X3 @ S4 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_4699_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B6: A,A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A10 )
=> ( ord_less @ A @ X6 @ B6 ) )
=> ( ( P @ A10 )
=> ( P @ ( insert @ A @ B6 @ A10 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_4700_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B6: A,A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A10 )
=> ( ord_less @ A @ B6 @ X6 ) )
=> ( ( P @ A10 )
=> ( P @ ( insert @ A @ B6 @ A10 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_4701_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_4702_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_4703_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_4704_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ B,T6: set @ C,S2: set @ B,I: C > B,J: B > C,T5: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) )
=> ( member @ C @ ( J @ A6 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T6 ) ) )
=> ( ! [B6: C] :
( ( member @ C @ B6 @ ( minus_minus @ ( set @ C ) @ T5 @ T6 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: C] :
( ( member @ C @ B6 @ ( minus_minus @ ( set @ C ) @ T5 @ T6 ) )
=> ( member @ B @ ( I @ B6 ) @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S5 )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B6: C] :
( ( member @ C @ B6 @ T6 )
=> ( ( H2 @ B6 )
= ( zero_zero @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S2 )
=> ( ( H2 @ ( J @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_4705_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_4706_prod__dvd__prod__subset2,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B3: set @ B,A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ B3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ A3 )
=> ( dvd_dvd @ A @ ( F2 @ A6 ) @ ( G @ A6 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_4707_prod__dvd__prod__subset,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B3: set @ B,A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B3 ) ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_4708_sum__eq__Suc0__iff,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ( F2 @ X2 )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y4: A] :
( ( member @ A @ Y4 @ A3 )
=> ( ( X2 != Y4 )
=> ( ( F2 @ Y4 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_4709_sum__eq__1__iff,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 )
= ( one_one @ nat ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ( F2 @ X2 )
= ( one_one @ nat ) )
& ! [Y4: A] :
( ( member @ A @ Y4 @ A3 )
=> ( ( X2 != Y4 )
=> ( ( F2 @ Y4 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_4710_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S )
=> ( ( F2 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_4711_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,B3: A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= B3 )
=> ( ( member @ B @ I @ S )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ B3 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_4712_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( set_or7035219750837199246ssThan @ nat @ I @ ( plus_plus @ nat @ J @ K ) )
= ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_4713_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,B3: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B3 ) @ ( G @ X2 ) @ ( zero_zero @ A ) )
@ A3 ) ) ) ) ).
% sum.inter_restrict
thf(fact_4714_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [X2: B] :
( ( G @ X2 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_4715_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,B3: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B3 ) @ ( G @ X2 ) @ ( one_one @ A ) )
@ A3 ) ) ) ) ).
% prod.inter_restrict
thf(fact_4716_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_4717_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [X2: B] :
( ( G @ X2 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_4718_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N4: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N4 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N4 )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( sums @ A @ F2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N4 ) ) ) ) ) ).
% sums_finite
thf(fact_4719_sums__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P ) )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( collect @ nat @ P ) ) ) ) ) ).
% sums_If_finite
thf(fact_4720_sums__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A3: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ A3 )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A3 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A3 ) ) ) ) ).
% sums_If_finite_set
thf(fact_4721_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N4: set @ nat,F2: nat > A] :
( ( finite_finite2 @ nat @ N4 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N4 )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A @ F2 )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N4 ) ) ) ) ) ).
% suminf_finite
thf(fact_4722_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K3: A] :
( ( member @ A @ K3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K3 ) @ A2 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_4723_subset__eq__atLeast0__atMost__finite,axiom,
! [N4: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N4 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_4724_exp__sum,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ B )
& ( real_Vector_banach @ B )
& ( real_V2822296259951069270ebra_1 @ B ) )
=> ! [I6: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ I6 ) )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X2: A] : ( exp @ B @ ( F2 @ X2 ) )
@ I6 ) ) ) ) ).
% exp_sum
thf(fact_4725_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D4 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D4 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4726_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D4 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D4 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_4727_ivl__disj__un__two__touch_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U2 ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_4728_sum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,K: nat] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_4729_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_4730_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_4731_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_4732_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_4733_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_4734_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A5: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B5 ) @ ( insert @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_4735_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_4736_sum_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ N ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% sum.last_plus
thf(fact_4737_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,I: B,F2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( member @ B @ I @ I6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_4738_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_4739_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ).
% sum_pos
thf(fact_4740_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,I: A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( member @ A @ I @ I6 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_4741_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_4742_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C3: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A3 ) )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B6: B] :
( ( member @ B @ B6 @ ( minus_minus @ ( set @ B ) @ C3 @ B3 ) )
=> ( ( H2 @ B6 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B3 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C3 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_4743_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C3: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A3 ) )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B6: B] :
( ( member @ B @ B6 @ ( minus_minus @ ( set @ B ) @ C3 @ B3 ) )
=> ( ( H2 @ B6 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C3 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B3 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_4744_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S2 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ T5 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_4745_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S2 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T5 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_4746_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S2: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S2 ) )
=> ( ( H2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T5 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_4747_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S2 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S2 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_4748_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B3 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_4749_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,B3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B3 ) ) ) ) ) ) ).
% sum_diff
thf(fact_4750_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S2: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T5 @ S2 ) )
=> ( ( H2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ S2 @ T5 ) )
=> ( ( G @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S2 @ T5 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T5 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_4751_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B3: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_4752_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C3: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A3 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B6: B] :
( ( member @ B @ B6 @ ( minus_minus @ ( set @ B ) @ C3 @ B3 ) )
=> ( ( H2 @ B6 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B3 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C3 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_4753_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C3: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A3 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B6: B] :
( ( member @ B @ B6 @ ( minus_minus @ ( set @ B ) @ C3 @ B3 ) )
=> ( ( H2 @ B6 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C3 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B3 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_4754_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S2 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_4755_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S2 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_4756_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S2: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S2 ) )
=> ( ( H2 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T5 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_4757_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S2 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_4758_infinite__imp__bij__betw2,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [H4: A > A] : ( bij_betw @ A @ A @ H4 @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_4759_infinite__imp__bij__betw,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [H4: A > A] : ( bij_betw @ A @ A @ H4 @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_4760_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
= ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_4761_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I4 ) ) @ ( F2 @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_4762_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_4763_sum__diff__nat,axiom,
! [A: $tType,B3: set @ A,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B3 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_4764_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( A2 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_4765_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_4766_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( A2 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_4767_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M3 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M3 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_4768_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M3 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M3 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_4769_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
@ ^ [Z7: A] :
( ( power_power @ A @ Z7 @ N )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_4770_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_4771_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_4772_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ B,T6: set @ C,H2: B > C,S2: set @ B,T5: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S5 )
=> ( ( G @ ( H2 @ A6 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B6: C] :
( ( member @ C @ B6 @ T6 )
=> ( ( G @ B6 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G @ ( H2 @ X2 ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T5 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_4773_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G: nat > A,S2: A,A3: set @ nat,S5: A,F2: nat > A] :
( ( sums @ A @ G @ S2 )
=> ( ( finite_finite2 @ nat @ A3 )
=> ( ( S5
= ( plus_plus @ A @ S2
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ A3 ) ) )
=> ( sums @ A
@ ^ [N3: nat] : ( if @ A @ ( member @ nat @ N3 @ A3 ) @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ S5 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_4774_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S5: set @ B,T6: set @ C,H2: B > C,S2: set @ B,T5: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S5 )
=> ( ( G @ ( H2 @ A6 ) )
= ( one_one @ A ) ) )
=> ( ! [B6: C] :
( ( member @ C @ B6 @ T6 )
=> ( ( G @ B6 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G @ ( H2 @ X2 ) )
@ S2 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T5 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_4775_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,P: B > $o,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H2 @ X2 ) @ ( G @ X2 ) )
@ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_4776_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,P: B > $o,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H2 @ X2 ) @ ( G @ X2 ) )
@ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_4777_ivl__disj__un__singleton_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U2 ) @ ( insert @ A @ U2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U2 ) ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_4778_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.head_if
thf(fact_4779_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_4780_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) )
& ( ord_less @ A @ ( F2 @ I2 ) @ ( G @ I2 ) ) ) )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_4781_even__prod__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_4782_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B3: set @ B,A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ B3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 )
=> ( ! [B6: B] :
( ( member @ B @ B6 @ ( minus_minus @ ( set @ B ) @ B3 @ A3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ B6 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B3 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_4783_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B3 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_4784_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_4785_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_4786_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,A2: B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ A2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_4787_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_4788_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_4789_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I6: set @ nat] :
( ( summable @ A @ F2 )
=> ( ( finite_finite2 @ nat @ I6 )
=> ( ! [N2: nat] :
( ( member @ nat @ N2 @ ( uminus_uminus @ ( set @ nat ) @ I6 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ I6 ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_4790_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_4791_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_4792_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( insert @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M @ ( pred_numeral @ K ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_4793_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_4794_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_4795_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: set @ B,F2: B > A,B2: A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ B2 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A5: B] : ( divide_divide @ A @ ( F2 @ A5 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [A5: B] : ( dvd_dvd @ A @ B2 @ ( F2 @ A5 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F2
@ ( inf_inf @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [A5: B] :
~ ( dvd_dvd @ A @ B2 @ ( F2 @ A5 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_4796_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A2: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_4797_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A2: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_4798_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_4799_summable__Cauchy,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ( ( summable @ A )
= ( ^ [F3: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [N7: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ N7 @ M3 )
=> ! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M3 @ N3 ) ) ) @ E3 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_4800_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B3: set @ A,A3: set @ A,B2: A,F2: A > B] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F2 @ B2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B3 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X3 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ B3 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_4801_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B3: set @ A,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ! [B6: A] :
( ( member @ A @ B6 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B6 ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ A6 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ B3 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_4802_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A3: set @ C,F2: C > B] :
( ( member @ C @ I @ A3 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A3 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X3 ) ) )
=> ( ( finite_finite2 @ C @ A3 )
=> ( ord_less_eq @ B @ ( F2 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F2 @ A3 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_4803_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A3: set @ B,B3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) )
=> ( ( F2 @ X3 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_4804_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A3: set @ B,F2: B > A,A2: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A2 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_4805_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A,S: A,K: nat] :
( ( sums @ A @ F2 @ S )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( sums @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N3 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ K ) @ K ) ) )
@ S ) ) ) ) ).
% sums_group
thf(fact_4806_ln__prod,axiom,
! [A: $tType,I6: set @ A,F2: A > real] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ I2 ) ) )
=> ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F2 @ I6 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_4807_even__set__encode__iff,axiom,
! [A3: set @ nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A3 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) ) ) ).
% even_set_encode_iff
thf(fact_4808_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A5: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ K3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% take_bit_sum
thf(fact_4809_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ X2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
& ( ( C2 @ I4 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_4810_polyfun__roots__finite,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z7: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z7 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_4811_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_4812_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N @ K ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_split
thf(fact_4813_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_4814_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_4815_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_4816_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_4817_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_4818_sum__power2,axiom,
! [K: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) )
= ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ ( one_one @ nat ) ) ) ).
% sum_power2
thf(fact_4819_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: nat > A,B2: nat > A] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( A2 @ I2 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( B2 @ J2 ) @ ( B2 @ I2 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ K3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_4820_finite__Diff__insert,axiom,
! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% finite_Diff_insert
thf(fact_4821_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A2: nat > nat,B2: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( A2 @ I2 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( B2 @ J2 ) @ ( B2 @ I2 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( A2 @ I4 ) @ ( B2 @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_4822_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_4823_finite__Collect__conjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_4824_finite__interval__int1,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A2 @ I4 )
& ( ord_less_eq @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_4825_finite__interval__int4,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A2 @ I4 )
& ( ord_less @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_4826_finite__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) )
= ( finite_finite2 @ A @ A3 ) ) ).
% finite_insert
thf(fact_4827_finite__Collect__subsets,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_4828_finite__interval__int2,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A2 @ I4 )
& ( ord_less @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_4829_finite__interval__int3,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A2 @ I4 )
& ( ord_less_eq @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_4830_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less @ nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_4831_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_4832_finite__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ complex
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_4833_finite__atLeastZeroLessThan__int,axiom,
! [U2: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U2 ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_4834_finite__divisors__int,axiom,
! [I: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( finite_finite2 @ int
@ ( collect @ int
@ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_4835_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R2: A > B > $o] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ? [Xa2: B] :
( ( member @ B @ Xa2 @ B3 )
& ( R2 @ X3 @ Xa2 ) ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ B3 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A3 )
& ( R2 @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_4836_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X_12: A] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_4837_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( ord_less_eq @ A @ X3 @ A2 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_4838_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( ord_less_eq @ A @ A2 @ X3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_4839_finite__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_subset
thf(fact_4840_infinite__super,axiom,
! [A: $tType,S2: set @ A,T5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ T5 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ T5 ) ) ) ).
% infinite_super
thf(fact_4841_rev__finite__subset,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_4842_finite_OinsertI,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) ) ) ).
% finite.insertI
thf(fact_4843_finite__psubset__induct,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ! [B11: set @ A] :
( ( ord_less @ ( set @ A ) @ B11 @ A10 )
=> ( P @ B11 ) )
=> ( P @ A10 ) ) )
=> ( P @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_4844_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_4845_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_4846_finite_Ocases,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A10: set @ A] :
( ? [A6: A] :
( A2
= ( insert @ A @ A6 @ A10 ) )
=> ~ ( finite_finite2 @ A @ A10 ) ) ) ) ).
% finite.cases
thf(fact_4847_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A5: set @ A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ? [A4: set @ A,B5: A] :
( ( A5
= ( insert @ A @ B5 @ A4 ) )
& ( finite_finite2 @ A @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_4848_finite__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X3 @ F6 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_4849_finite__ne__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( F4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] : ( P @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( F6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X3 @ F6 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_4850_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] :
( ! [A10: set @ A] :
( ~ ( finite_finite2 @ A @ A10 )
=> ( P @ A10 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X3 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ X3 @ F6 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_4851_finite__subset__induct_H,axiom,
! [A: $tType,F4: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A6 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ F6 @ A3 )
=> ( ~ ( member @ A @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ A6 @ F6 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_4852_finite__subset__induct,axiom,
! [A: $tType,F4: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A6 @ A3 )
=> ( ~ ( member @ A @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert @ A @ A6 @ F6 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_4853_infinite__remove,axiom,
! [A: $tType,S2: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_4854_infinite__coinduct,axiom,
! [A: $tType,X5: ( set @ A ) > $o,A3: set @ A] :
( ( X5 @ A3 )
=> ( ! [A10: set @ A] :
( ( X5 @ A10 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A10 )
& ( ( X5 @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_4855_finite__empty__induct,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ A3 )
=> ( ! [A6: A,A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ( member @ A @ A6 @ A10 )
=> ( ( P @ A10 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_4856_finite__remove__induct,axiom,
! [A: $tType,B3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ( A10
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A10 @ B3 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A10 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_4857_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B3: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B3 )
=> ( P @ B3 ) )
=> ( ! [A10: set @ A] :
( ( finite_finite2 @ A @ A10 )
=> ( ( A10
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A10 @ B3 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A10 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A10 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_4858_finite__induct__select,axiom,
! [A: $tType,S2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T7: set @ A] :
( ( ord_less @ ( set @ A ) @ T7 @ S2 )
=> ( ( P @ T7 )
=> ? [X6: A] :
( ( member @ A @ X6 @ ( minus_minus @ ( set @ A ) @ S2 @ T7 ) )
& ( P @ ( insert @ A @ X6 @ T7 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_4859_finite__nat__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S6: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S6 @ ( set_ord_atMost @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_4860_finite__nat__bounded,axiom,
! [S2: set @ nat] :
( ( finite_finite2 @ nat @ S2 )
=> ? [K2: nat] : ( ord_less_eq @ ( set @ nat ) @ S2 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_4861_finite__nat__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S6: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S6 @ ( set_ord_lessThan @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_4862_infinite__int__iff__unbounded__le,axiom,
! [S2: set @ int] :
( ( ~ ( finite_finite2 @ int @ S2 ) )
= ( ! [M3: int] :
? [N3: int] :
( ( ord_less_eq @ int @ M3 @ ( abs_abs @ int @ N3 ) )
& ( member @ int @ N3 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_4863_infinite__int__iff__unbounded,axiom,
! [S2: set @ int] :
( ( ~ ( finite_finite2 @ int @ S2 ) )
= ( ! [M3: int] :
? [N3: int] :
( ( ord_less @ int @ M3 @ ( abs_abs @ int @ N3 ) )
& ( member @ int @ N3 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_4864_infinite__nat__iff__unbounded,axiom,
! [S2: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S2 ) )
= ( ! [M3: nat] :
? [N3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ( member @ nat @ N3 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_4865_unbounded__k__infinite,axiom,
! [K: nat,S2: set @ nat] :
( ! [M4: nat] :
( ( ord_less @ nat @ K @ M4 )
=> ? [N9: nat] :
( ( ord_less @ nat @ M4 @ N9 )
& ( member @ nat @ N9 @ S2 ) ) )
=> ~ ( finite_finite2 @ nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_4866_infinite__nat__iff__unbounded__le,axiom,
! [S2: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S2 ) )
= ( ! [M3: nat] :
? [N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
& ( member @ nat @ N3 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_4867_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F4: set @ A,I6: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I6 )
& ( ( F2 @ I4 )
!= ( zero_zero @ B ) ) ) )
@ F4 )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_4868_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
( A4
= ( insert @ A @ ( the_elem @ A @ A4 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_4869_Code__Target__Int_Opositive__def,axiom,
( code_Target_positive
= ( numeral_numeral @ int ) ) ).
% Code_Target_Int.positive_def
thf(fact_4870_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P5: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_4871_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_4872_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P5: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( P5 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I6 ) ) )
& ( ~ ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
= ( plus_plus @ A @ ( P5 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I6 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_4873_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,I6: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( G @ X2 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) ) ) ).
% sum.non_neutral'
thf(fact_4874_is__singletonI_H,axiom,
! [A: $tType,A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y3 @ A3 )
=> ( X3 = Y3 ) ) )
=> ( is_singleton @ A @ A3 ) ) ) ).
% is_singletonI'
thf(fact_4875_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G @ I4 ) @ ( H2 @ I4 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_4876_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T5: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S2 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T5 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_4877_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T5: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T5 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T5 @ S2 ) )
=> ( ( H2 @ I2 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S2 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T5 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_4878_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T5: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S2 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T5 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_4879_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T5: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T5 @ S2 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S2 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ T5 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_4880_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( G @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( H2 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G @ I4 ) @ ( H2 @ I4 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_4881_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P6: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I7 )
& ( ( P6 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P6
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I7 )
& ( ( P6 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_4882_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
? [X2: A] :
( A4
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_4883_is__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( is_singleton @ A @ A3 )
=> ~ ! [X3: A] :
( A3
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_4884_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I6: set @ A,F2: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I6 )
& ( ( F2 @ I4 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_4885_csqrt_Osimps_I1_J,axiom,
! [Z3: complex] :
( ( re @ ( csqrt @ Z3 ) )
= ( sqrt @ ( divide_divide @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% csqrt.simps(1)
thf(fact_4886_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A5: B] :
( ( member @ B @ A5 @ A3 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ A5 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_4887_insert_H__bound__height,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ X ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).
% insert'_bound_height
thf(fact_4888_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_4889_complex__Re__of__nat,axiom,
! [N: nat] :
( ( re @ ( semiring_1_of_nat @ complex @ N ) )
= ( semiring_1_of_nat @ real @ N ) ) ).
% complex_Re_of_nat
thf(fact_4890_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less_eq @ nat @ I4 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_4891_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_4892_card_Oinfinite,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_4893_complex__Re__numeral,axiom,
! [V2: num] :
( ( re @ ( numeral_numeral @ complex @ V2 ) )
= ( numeral_numeral @ real @ V2 ) ) ).
% complex_Re_numeral
thf(fact_4894_prod__constant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : Y
@ A3 )
= ( power_power @ A @ Y @ ( finite_card @ B @ A3 ) ) ) ) ).
% prod_constant
thf(fact_4895_card__0__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_4896_card__insert__disjoint,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A3 ) )
= ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_4897_Re__sum,axiom,
! [A: $tType,F2: A > complex,S: set @ A] :
( ( re @ ( groups7311177749621191930dd_sum @ A @ complex @ F2 @ S ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X2: A] : ( re @ ( F2 @ X2 ) )
@ S ) ) ).
% Re_sum
thf(fact_4898_Re__divide__of__nat,axiom,
! [Z3: complex,N: nat] :
( ( re @ ( divide_divide @ complex @ Z3 @ ( semiring_1_of_nat @ complex @ N ) ) )
= ( divide_divide @ real @ ( re @ Z3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ).
% Re_divide_of_nat
thf(fact_4899_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y: A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : Y
@ A3 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ Y ) ) ) ).
% sum_constant
thf(fact_4900_card__Diff__insert,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ A2 @ B3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_4901_Re__divide__numeral,axiom,
! [Z3: complex,W2: num] :
( ( re @ ( divide_divide @ complex @ Z3 @ ( numeral_numeral @ complex @ W2 ) ) )
= ( divide_divide @ real @ ( re @ Z3 ) @ ( numeral_numeral @ real @ W2 ) ) ) ).
% Re_divide_numeral
thf(fact_4902_cos__Arg__i__mult__zero,axiom,
! [Y: complex] :
( ( Y
!= ( zero_zero @ complex ) )
=> ( ( ( re @ Y )
= ( zero_zero @ real ) )
=> ( ( cos @ real @ ( arg @ Y ) )
= ( zero_zero @ real ) ) ) ) ).
% cos_Arg_i_mult_zero
thf(fact_4903_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,P: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( zero_neq_one_of_bool @ A @ ( P @ X2 ) )
@ A3 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_4904_sums__Re,axiom,
! [X5: nat > complex,A2: complex] :
( ( sums @ complex @ X5 @ A2 )
=> ( sums @ real
@ ^ [N3: nat] : ( re @ ( X5 @ N3 ) )
@ ( re @ A2 ) ) ) ).
% sums_Re
thf(fact_4905_n__subsets,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A3 )
& ( ( finite_card @ A @ B4 )
= K ) ) ) )
= ( binomial @ ( finite_card @ A @ A3 ) @ K ) ) ) ).
% n_subsets
thf(fact_4906_card__subset__eq,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ( finite_card @ A @ A3 )
= ( finite_card @ A @ B3 ) )
=> ( A3 = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_4907_infinite__arbitrarily__large,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [B7: set @ A] :
( ( finite_finite2 @ A @ B7 )
& ( ( finite_card @ A @ B7 )
= N )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_4908_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B3: set @ A,A3: set @ B,R3: B > A > $o] :
( ( finite_finite2 @ A @ B3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ A3 )
=> ? [B12: A] :
( ( member @ A @ B12 @ B3 )
& ( R3 @ A6 @ B12 ) ) )
=> ( ! [A1: B,A22: B,B6: A] :
( ( member @ B @ A1 @ A3 )
=> ( ( member @ B @ A22 @ A3 )
=> ( ( member @ A @ B6 @ B3 )
=> ( ( R3 @ A1 @ B6 )
=> ( ( R3 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_4909_card__insert__le,axiom,
! [A: $tType,A3: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ ( insert @ A @ X @ A3 ) ) ) ).
% card_insert_le
thf(fact_4910_Cauchy__Re,axiom,
! [X5: nat > complex] :
( ( topolo3814608138187158403Cauchy @ complex @ X5 )
=> ( topolo3814608138187158403Cauchy @ real
@ ^ [N3: nat] : ( re @ ( X5 @ N3 ) ) ) ) ).
% Cauchy_Re
thf(fact_4911_imaginary__unit_Osimps_I1_J,axiom,
( ( re @ imaginary_unit )
= ( zero_zero @ real ) ) ).
% imaginary_unit.simps(1)
thf(fact_4912_complex__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( re @ X ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% complex_Re_le_cmod
thf(fact_4913_zero__complex_Osimps_I1_J,axiom,
( ( re @ ( zero_zero @ complex ) )
= ( zero_zero @ real ) ) ).
% zero_complex.simps(1)
thf(fact_4914_sum__multicount__gen,axiom,
! [A: $tType,B: $tType,S: set @ A,T2: set @ B,R2: A > B > $o,K: B > nat] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T2 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ S )
& ( R2 @ I4 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T2 )
& ( R2 @ I4 @ J3 ) ) ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ nat @ K @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_4915_card__eq__sum,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( one_one @ nat ) ) ) ).
% card_eq_sum
thf(fact_4916_card__2__iff_H,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S2 )
& ? [Y4: A] :
( ( member @ A @ Y4 @ S2 )
& ( X2 != Y4 )
& ! [Z7: A] :
( ( member @ A @ Z7 @ S2 )
=> ( ( Z7 = X2 )
| ( Z7 = Y4 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_4917_card__eq__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% card_eq_0_iff
thf(fact_4918_card__ge__0__finite,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
=> ( finite_finite2 @ A @ A3 ) ) ).
% card_ge_0_finite
thf(fact_4919_card__Suc__eq__finite,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
= ( ? [B5: A,B4: set @ A] :
( ( A3
= ( insert @ A @ B5 @ B4 ) )
& ~ ( member @ A @ B5 @ B4 )
& ( ( finite_card @ A @ B4 )
= K )
& ( finite_finite2 @ A @ B4 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_4920_card__insert__if,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A3 ) )
= ( finite_card @ A @ A3 ) ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A3 ) )
= ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_4921_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S2: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S2 ) )
=> ~ ! [T7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T7 @ S2 )
=> ( ( ( finite_card @ A @ T7 )
= N )
=> ~ ( finite_finite2 @ A @ T7 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_4922_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F4: set @ A,C3: nat] :
( ! [G3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G3 @ F4 )
=> ( ( finite_finite2 @ A @ G3 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G3 ) @ C3 ) ) )
=> ( ( finite_finite2 @ A @ F4 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F4 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_4923_card__seteq,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B3 ) @ ( finite_card @ A @ A3 ) )
=> ( A3 = B3 ) ) ) ) ).
% card_seteq
thf(fact_4924_card__mono,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% card_mono
thf(fact_4925_card__1__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( one_one @ nat ) )
=> ~ ! [X3: A] :
( A3
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_4926_card__less__sym__Diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_4927_card__le__sym__Diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_4928_card__Un__le,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ).
% card_Un_le
thf(fact_4929_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H4: A > nat] : ( bij_betw @ A @ nat @ H4 @ M7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_4930_psubset__card__mono,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_4931_summable__Re,axiom,
! [F2: nat > complex] :
( ( summable @ complex @ F2 )
=> ( summable @ real
@ ^ [X2: nat] : ( re @ ( F2 @ X2 ) ) ) ) ).
% summable_Re
thf(fact_4932_abs__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( re @ X ) ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% abs_Re_le_cmod
thf(fact_4933_Re__csqrt,axiom,
! [Z3: complex] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( csqrt @ Z3 ) ) ) ).
% Re_csqrt
thf(fact_4934_card__less,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_4935_card__less__Suc,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_4936_card__less__Suc2,axiom,
! [M7: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_4937_card__atLeastZeroLessThan__int,axiom,
! [U2: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U2 ) )
= ( nat2 @ U2 ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_4938_sum__constant__scaleR,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Y: A,A3: set @ C] :
( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X2: C] : Y
@ A3 )
= ( real_V8093663219630862766scaleR @ A @ ( semiring_1_of_nat @ real @ ( finite_card @ C @ A3 ) ) @ Y ) ) ) ).
% sum_constant_scaleR
thf(fact_4939_sum__Suc,axiom,
! [A: $tType,F2: A > nat,A3: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( suc @ ( F2 @ X2 ) )
@ A3 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( finite_card @ A @ A3 ) ) ) ).
% sum_Suc
thf(fact_4940_subset__card__intvl__is__intvl,axiom,
! [A3: set @ nat,K: nat] :
( ( ord_less_eq @ ( set @ nat ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A3 ) ) ) )
=> ( A3
= ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A3 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_4941_sum__multicount,axiom,
! [A: $tType,B: $tType,S2: set @ A,T5: set @ B,R2: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ B @ T5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T5 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ S2 )
& ( R2 @ I4 @ X3 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T5 )
& ( R2 @ I4 @ J3 ) ) ) )
@ S2 )
= ( times_times @ nat @ K @ ( finite_card @ B @ T5 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_4942_real__of__card,axiom,
! [A: $tType,A3: set @ A] :
( ( semiring_1_of_nat @ real @ ( finite_card @ A @ A3 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X2: A] : ( one_one @ real )
@ A3 ) ) ).
% real_of_card
thf(fact_4943_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F2: B > A,K6: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ K6 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K6 ) ) ) ) ).
% sum_bounded_above
thf(fact_4944_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,K6: A,F2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ K6 @ ( F2 @ I2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ).
% sum_bounded_below
thf(fact_4945_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ A3 )
=> ( X2 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_4946_card__1__singleton__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( A3
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_4947_card__eq__SucD,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
=> ? [B6: A,B7: set @ A] :
( ( A3
= ( insert @ A @ B6 @ B7 ) )
& ~ ( member @ A @ B6 @ B7 )
& ( ( finite_card @ A @ B7 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B7
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_4948_card__Suc__eq,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K ) )
= ( ? [B5: A,B4: set @ A] :
( ( A3
= ( insert @ A @ B5 @ B4 ) )
& ~ ( member @ A @ B5 @ B4 )
& ( ( finite_card @ A @ B4 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_4949_card__gt__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
= ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A3 ) ) ) ).
% card_gt_0_iff
thf(fact_4950_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A3: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A3 ) )
= ( ? [A5: A,B4: set @ A] :
( ( A3
= ( insert @ A @ A5 @ B4 ) )
& ~ ( member @ A @ A5 @ B4 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B4 ) )
& ( finite_finite2 @ A @ B4 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_4951_card__Diff1__le,axiom,
! [A: $tType,A3: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ).
% card_Diff1_le
thf(fact_4952_card__Diff__subset,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_4953_card__psubset,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) )
=> ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% card_psubset
thf(fact_4954_diff__card__le__card__Diff,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_4955_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H4: nat > A] : ( bij_betw @ nat @ A @ H4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_4956_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
@ ^ [Z7: A] :
( ( power_power @ A @ Z7 @ N )
= ( one_one @ A ) ) ) )
@ N ) ) ) ).
% card_roots_unity
thf(fact_4957_subset__eq__atLeast0__lessThan__card,axiom,
! [N4: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N4 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_4958_card__sum__le__nat__sum,axiom,
! [S2: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S2 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ S2 ) ) ).
% card_sum_le_nat_sum
thf(fact_4959_card__nth__roots,axiom,
! [C2: complex,N: nat] :
( ( C2
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= C2 ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_4960_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z7: complex] :
( ( power_power @ complex @ Z7 @ N )
= ( one_one @ complex ) ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_4961_card__2__iff,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X2: A,Y4: A] :
( ( S2
= ( insert @ A @ X2 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X2 != Y4 ) ) ) ) ).
% card_2_iff
thf(fact_4962_card__3__iff,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X2: A,Y4: A,Z7: A] :
( ( S2
= ( insert @ A @ X2 @ ( insert @ A @ Y4 @ ( insert @ A @ Z7 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X2 != Y4 )
& ( Y4 != Z7 )
& ( X2 != Z7 ) ) ) ) ).
% card_3_iff
thf(fact_4963_odd__card__imp__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A3 ) )
=> ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_4964_card__Suc__Diff1,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_4965_card_Oinsert__remove,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A3 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_4966_card_Oremove,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ A3 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_4967_card__Diff1__less__iff,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) )
= ( ( finite_finite2 @ A @ A3 )
& ( member @ A @ X @ A3 ) ) ) ).
% card_Diff1_less_iff
thf(fact_4968_card__Diff2__less,axiom,
! [A: $tType,A3: set @ A,X: A,Y: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_4969_card__Diff1__less,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ).
% card_Diff1_less
thf(fact_4970_card__Diff__singleton,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_4971_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_4972_sum__norm__bound,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S2: set @ B,F2: B > A,K6: real] :
( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X3 ) ) @ K6 ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S2 ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( finite_card @ B @ S2 ) ) @ K6 ) ) ) ) ).
% sum_norm_bound
thf(fact_4973_cmod__plus__Re__le__0__iff,axiom,
! [Z3: complex] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( re @ Z3 ) ) @ ( zero_zero @ real ) )
= ( ( re @ Z3 )
= ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) ) ) ) ).
% cmod_plus_Re_le_0_iff
thf(fact_4974_cos__n__Re__cis__pow__n,axiom,
! [N: nat,A2: real] :
( ( cos @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A2 ) )
= ( re @ ( power_power @ complex @ ( cis @ A2 ) @ N ) ) ) ).
% cos_n_Re_cis_pow_n
thf(fact_4975_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F2: B > A,N: A,K: nat] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) )
& ( ord_less_eq @ A @ ( F2 @ I2 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_4976_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F2: B > A,K6: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less @ A @ ( F2 @ I2 ) @ K6 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A3 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K6 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_4977_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A3: set @ B,F2: B > A,K6: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( divide_divide @ A @ K6 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ K6 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_4978_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_4979_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A2: B,B2: B > A,C2: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C2 )
@ S2 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S2 ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C2 )
@ S2 )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S2 ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_4980_polyfun__roots__card,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z7: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z7 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ).
% polyfun_roots_card
thf(fact_4981_insertsimp_H,axiom,
! [T2: vEBT_VEBT,N: nat,L: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_VEBT_minNull @ T2 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ L ) @ ( one_one @ nat ) ) ) ) ).
% insertsimp'
thf(fact_4982_polyfun__rootbound,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z7: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z7 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z7: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z7 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_4983_csqrt_Ocode,axiom,
( csqrt
= ( ^ [Z7: complex] :
( complex2 @ ( sqrt @ ( divide_divide @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z7 ) @ ( re @ Z7 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( times_times @ real
@ ( if @ real
@ ( ( im @ Z7 )
= ( zero_zero @ real ) )
@ ( one_one @ real )
@ ( sgn_sgn @ real @ ( im @ Z7 ) ) )
@ ( sqrt @ ( divide_divide @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z7 ) @ ( re @ Z7 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% csqrt.code
thf(fact_4984_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_4985_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_4986_complex__Im__fact,axiom,
! [N: nat] :
( ( im @ ( semiring_char_0_fact @ complex @ N ) )
= ( zero_zero @ real ) ) ).
% complex_Im_fact
thf(fact_4987_complex__Im__of__int,axiom,
! [Z3: int] :
( ( im @ ( ring_1_of_int @ complex @ Z3 ) )
= ( zero_zero @ real ) ) ).
% complex_Im_of_int
thf(fact_4988_Im__complex__of__real,axiom,
! [Z3: real] :
( ( im @ ( real_Vector_of_real @ complex @ Z3 ) )
= ( zero_zero @ real ) ) ).
% Im_complex_of_real
thf(fact_4989_Im__power__real,axiom,
! [X: complex,N: nat] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( im @ ( power_power @ complex @ X @ N ) )
= ( zero_zero @ real ) ) ) ).
% Im_power_real
thf(fact_4990_complex__Im__numeral,axiom,
! [V2: num] :
( ( im @ ( numeral_numeral @ complex @ V2 ) )
= ( zero_zero @ real ) ) ).
% complex_Im_numeral
thf(fact_4991_complex__Im__of__nat,axiom,
! [N: nat] :
( ( im @ ( semiring_1_of_nat @ complex @ N ) )
= ( zero_zero @ real ) ) ).
% complex_Im_of_nat
thf(fact_4992_Im__sum,axiom,
! [A: $tType,F2: A > complex,S: set @ A] :
( ( im @ ( groups7311177749621191930dd_sum @ A @ complex @ F2 @ S ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X2: A] : ( im @ ( F2 @ X2 ) )
@ S ) ) ).
% Im_sum
thf(fact_4993_Re__power__real,axiom,
! [X: complex,N: nat] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( re @ ( power_power @ complex @ X @ N ) )
= ( power_power @ real @ ( re @ X ) @ N ) ) ) ).
% Re_power_real
thf(fact_4994_Im__divide__numeral,axiom,
! [Z3: complex,W2: num] :
( ( im @ ( divide_divide @ complex @ Z3 @ ( numeral_numeral @ complex @ W2 ) ) )
= ( divide_divide @ real @ ( im @ Z3 ) @ ( numeral_numeral @ real @ W2 ) ) ) ).
% Im_divide_numeral
thf(fact_4995_Im__divide__of__nat,axiom,
! [Z3: complex,N: nat] :
( ( im @ ( divide_divide @ complex @ Z3 @ ( semiring_1_of_nat @ complex @ N ) ) )
= ( divide_divide @ real @ ( im @ Z3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ).
% Im_divide_of_nat
thf(fact_4996_csqrt__of__real__nonneg,axiom,
! [X: complex] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ X ) )
=> ( ( csqrt @ X )
= ( real_Vector_of_real @ complex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% csqrt_of_real_nonneg
thf(fact_4997_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_4998_sums__Im,axiom,
! [X5: nat > complex,A2: complex] :
( ( sums @ complex @ X5 @ A2 )
=> ( sums @ real
@ ^ [N3: nat] : ( im @ ( X5 @ N3 ) )
@ ( im @ A2 ) ) ) ).
% sums_Im
thf(fact_4999_Cauchy__Im,axiom,
! [X5: nat > complex] :
( ( topolo3814608138187158403Cauchy @ complex @ X5 )
=> ( topolo3814608138187158403Cauchy @ real
@ ^ [N3: nat] : ( im @ ( X5 @ N3 ) ) ) ) ).
% Cauchy_Im
thf(fact_5000_zero__complex_Osimps_I2_J,axiom,
( ( im @ ( zero_zero @ complex ) )
= ( zero_zero @ real ) ) ).
% zero_complex.simps(2)
thf(fact_5001_one__complex_Osimps_I2_J,axiom,
( ( im @ ( one_one @ complex ) )
= ( zero_zero @ real ) ) ).
% one_complex.simps(2)
thf(fact_5002_sums__complex__iff,axiom,
( ( sums @ complex )
= ( ^ [F3: nat > complex,X2: complex] :
( ( sums @ real
@ ^ [Y4: nat] : ( re @ ( F3 @ Y4 ) )
@ ( re @ X2 ) )
& ( sums @ real
@ ^ [Y4: nat] : ( im @ ( F3 @ Y4 ) )
@ ( im @ X2 ) ) ) ) ) ).
% sums_complex_iff
thf(fact_5003_summable__Im,axiom,
! [F2: nat > complex] :
( ( summable @ complex @ F2 )
=> ( summable @ real
@ ^ [X2: nat] : ( im @ ( F2 @ X2 ) ) ) ) ).
% summable_Im
thf(fact_5004_complex__is__Int__iff,axiom,
! [Z3: complex] :
( ( member @ complex @ Z3 @ ( ring_1_Ints @ complex ) )
= ( ( ( im @ Z3 )
= ( zero_zero @ real ) )
& ? [I4: int] :
( ( re @ Z3 )
= ( ring_1_of_int @ real @ I4 ) ) ) ) ).
% complex_is_Int_iff
thf(fact_5005_abs__Im__le__cmod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( im @ X ) ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% abs_Im_le_cmod
thf(fact_5006_summable__complex__iff,axiom,
( ( summable @ complex )
= ( ^ [F3: nat > complex] :
( ( summable @ real
@ ^ [X2: nat] : ( re @ ( F3 @ X2 ) ) )
& ( summable @ real
@ ^ [X2: nat] : ( im @ ( F3 @ X2 ) ) ) ) ) ) ).
% summable_complex_iff
thf(fact_5007_Im__eq__0,axiom,
! [Z3: complex] :
( ( ( abs_abs @ real @ ( re @ Z3 ) )
= ( real_V7770717601297561774m_norm @ complex @ Z3 ) )
=> ( ( im @ Z3 )
= ( zero_zero @ real ) ) ) ).
% Im_eq_0
thf(fact_5008_cmod__eq__Im,axiom,
! [Z3: complex] :
( ( ( re @ Z3 )
= ( zero_zero @ real ) )
=> ( ( real_V7770717601297561774m_norm @ complex @ Z3 )
= ( abs_abs @ real @ ( im @ Z3 ) ) ) ) ).
% cmod_eq_Im
thf(fact_5009_cmod__eq__Re,axiom,
! [Z3: complex] :
( ( ( im @ Z3 )
= ( zero_zero @ real ) )
=> ( ( real_V7770717601297561774m_norm @ complex @ Z3 )
= ( abs_abs @ real @ ( re @ Z3 ) ) ) ) ).
% cmod_eq_Re
thf(fact_5010_cmod__Im__le__iff,axiom,
! [X: complex,Y: complex] :
( ( ( re @ X )
= ( re @ Y ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ complex @ X ) @ ( real_V7770717601297561774m_norm @ complex @ Y ) )
= ( ord_less_eq @ real @ ( abs_abs @ real @ ( im @ X ) ) @ ( abs_abs @ real @ ( im @ Y ) ) ) ) ) ).
% cmod_Im_le_iff
thf(fact_5011_cmod__Re__le__iff,axiom,
! [X: complex,Y: complex] :
( ( ( im @ X )
= ( im @ Y ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ complex @ X ) @ ( real_V7770717601297561774m_norm @ complex @ Y ) )
= ( ord_less_eq @ real @ ( abs_abs @ real @ ( re @ X ) ) @ ( abs_abs @ real @ ( re @ Y ) ) ) ) ) ).
% cmod_Re_le_iff
thf(fact_5012_csqrt__principal,axiom,
! [Z3: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( csqrt @ Z3 ) ) )
| ( ( ( re @ ( csqrt @ Z3 ) )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( csqrt @ Z3 ) ) ) ) ) ).
% csqrt_principal
thf(fact_5013_cmod__le,axiom,
! [Z3: complex] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( plus_plus @ real @ ( abs_abs @ real @ ( re @ Z3 ) ) @ ( abs_abs @ real @ ( im @ Z3 ) ) ) ) ).
% cmod_le
thf(fact_5014_sin__n__Im__cis__pow__n,axiom,
! [N: nat,A2: real] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A2 ) )
= ( im @ ( power_power @ complex @ ( cis @ A2 ) @ N ) ) ) ).
% sin_n_Im_cis_pow_n
thf(fact_5015_cmod__power2,axiom,
! [Z3: complex] :
( ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ real @ ( power_power @ real @ ( re @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% cmod_power2
thf(fact_5016_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_5017_Re__power2,axiom,
! [X: complex] :
( ( re @ ( power_power @ complex @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ real @ ( power_power @ real @ ( re @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Re_power2
thf(fact_5018_complex__eq__0,axiom,
! [Z3: complex] :
( ( Z3
= ( zero_zero @ complex ) )
= ( ( plus_plus @ real @ ( power_power @ real @ ( re @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ) ).
% complex_eq_0
thf(fact_5019_norm__complex__def,axiom,
( ( real_V7770717601297561774m_norm @ complex )
= ( ^ [Z7: complex] : ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z7 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z7 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% norm_complex_def
thf(fact_5020_inverse__complex_Osimps_I1_J,axiom,
! [X: complex] :
( ( re @ ( inverse_inverse @ complex @ X ) )
= ( divide_divide @ real @ ( re @ X ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% inverse_complex.simps(1)
thf(fact_5021_complex__neq__0,axiom,
! [Z3: complex] :
( ( Z3
!= ( zero_zero @ complex ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_neq_0
thf(fact_5022_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_5023_csqrt__unique,axiom,
! [W2: complex,Z3: complex] :
( ( ( power_power @ complex @ W2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z3 )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ W2 ) )
| ( ( ( re @ W2 )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ W2 ) ) ) )
=> ( ( csqrt @ Z3 )
= W2 ) ) ) ).
% csqrt_unique
thf(fact_5024_csqrt__square,axiom,
! [B2: complex] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ B2 ) )
| ( ( ( re @ B2 )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ B2 ) ) ) )
=> ( ( csqrt @ ( power_power @ complex @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= B2 ) ) ).
% csqrt_square
thf(fact_5025_inverse__complex_Osimps_I2_J,axiom,
! [X: complex] :
( ( im @ ( inverse_inverse @ complex @ X ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( im @ X ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% inverse_complex.simps(2)
thf(fact_5026_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_5027_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_5028_complex__unit__circle,axiom,
! [Z3: complex] :
( ( Z3
!= ( zero_zero @ complex ) )
=> ( ( plus_plus @ real @ ( power_power @ real @ ( divide_divide @ real @ ( re @ Z3 ) @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( divide_divide @ real @ ( im @ Z3 ) @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ) ).
% complex_unit_circle
thf(fact_5029_inverse__complex_Ocode,axiom,
( ( inverse_inverse @ complex )
= ( ^ [X2: complex] : ( complex2 @ ( divide_divide @ real @ ( re @ X2 ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( uminus_uminus @ real @ ( im @ X2 ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% inverse_complex.code
thf(fact_5030_Complex__divide,axiom,
( ( divide_divide @ complex )
= ( ^ [X2: complex,Y4: complex] : ( complex2 @ ( divide_divide @ real @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X2 ) @ ( re @ Y4 ) ) @ ( times_times @ real @ ( im @ X2 ) @ ( 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 @ X2 ) @ ( re @ Y4 ) ) @ ( times_times @ real @ ( re @ X2 ) @ ( 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_5031_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_5032_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_5033_series__comparison__complex,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G: nat > complex,N4: nat,F2: nat > A] :
( ( summable @ complex @ G )
=> ( ! [N2: nat] : ( member @ complex @ ( G @ N2 ) @ ( real_Vector_Reals @ complex ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( G @ N2 ) ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ ( G @ N2 ) ) ) )
=> ( summable @ A @ F2 ) ) ) ) ) ) ).
% series_comparison_complex
thf(fact_5034_Reals__of__nat,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_of_nat
thf(fact_5035_Reals__0,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_0
thf(fact_5036_Reals__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A2: A,N: nat] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( power_power @ A @ A2 @ N ) @ ( real_Vector_Reals @ A ) ) ) ) ).
% Reals_power
thf(fact_5037_Reals__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W2: num] : ( member @ A @ ( numeral_numeral @ A @ W2 ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_numeral
thf(fact_5038_complex__is__Real__iff,axiom,
! [Z3: complex] :
( ( member @ complex @ Z3 @ ( real_Vector_Reals @ complex ) )
= ( ( im @ Z3 )
= ( zero_zero @ real ) ) ) ).
% complex_is_Real_iff
thf(fact_5039_nonzero__Reals__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( member @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ) ).
% nonzero_Reals_divide
thf(fact_5040_Complex__in__Reals,axiom,
! [X: real] : ( member @ complex @ ( complex2 @ X @ ( zero_zero @ real ) ) @ ( real_Vector_Reals @ complex ) ) ).
% Complex_in_Reals
thf(fact_5041_nonzero__Reals__inverse,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( member @ A @ ( inverse_inverse @ A @ A2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% nonzero_Reals_inverse
thf(fact_5042_Re__prod__Reals,axiom,
! [A: $tType,A3: set @ A,F2: A > complex] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ complex @ ( F2 @ X3 ) @ ( real_Vector_Reals @ complex ) ) )
=> ( ( re @ ( groups7121269368397514597t_prod @ A @ complex @ F2 @ A3 ) )
= ( groups7121269368397514597t_prod @ A @ real
@ ^ [X2: A] : ( re @ ( F2 @ X2 ) )
@ A3 ) ) ) ).
% Re_prod_Reals
thf(fact_5043_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_5044_card__lists__length__le,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A3 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_5045_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_5046_length__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ).
% length_map
thf(fact_5047_complex__cnj__numeral,axiom,
! [W2: num] :
( ( cnj @ ( numeral_numeral @ complex @ W2 ) )
= ( numeral_numeral @ complex @ W2 ) ) ).
% complex_cnj_numeral
thf(fact_5048_complex__cnj__of__nat,axiom,
! [N: nat] :
( ( cnj @ ( semiring_1_of_nat @ complex @ N ) )
= ( semiring_1_of_nat @ complex @ N ) ) ).
% complex_cnj_of_nat
thf(fact_5049_cnj__sum,axiom,
! [A: $tType,F2: A > complex,S: set @ A] :
( ( cnj @ ( groups7311177749621191930dd_sum @ A @ complex @ F2 @ S ) )
= ( groups7311177749621191930dd_sum @ A @ complex
@ ^ [X2: A] : ( cnj @ ( F2 @ X2 ) )
@ S ) ) ).
% cnj_sum
thf(fact_5050_cnj__prod,axiom,
! [A: $tType,F2: A > complex,S: set @ A] :
( ( cnj @ ( groups7121269368397514597t_prod @ A @ complex @ F2 @ S ) )
= ( groups7121269368397514597t_prod @ A @ complex
@ ^ [X2: A] : ( cnj @ ( F2 @ X2 ) )
@ S ) ) ).
% cnj_prod
thf(fact_5051_complex__cnj__neg__numeral,axiom,
! [W2: num] :
( ( cnj @ ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W2 ) ) )
= ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W2 ) ) ) ).
% complex_cnj_neg_numeral
thf(fact_5052_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_5053_finite__maxlen,axiom,
! [A: $tType,M7: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M7 )
=> ? [N2: nat] :
! [X6: list @ A] :
( ( member @ ( list @ A ) @ X6 @ M7 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X6 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_5054_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs3: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_5055_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_5056_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs3: list @ A] :
( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_5057_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_5058_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,Xs: list @ B,G: C > A,Ys: list @ C] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ C @ A @ G @ Ys ) )
=> ( ( size_size @ ( list @ B ) @ Xs )
= ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_5059_sums__cnj,axiom,
! [F2: nat > complex,L: complex] :
( ( sums @ complex
@ ^ [X2: nat] : ( cnj @ ( F2 @ X2 ) )
@ ( cnj @ L ) )
= ( sums @ complex @ F2 @ L ) ) ).
% sums_cnj
thf(fact_5060_length__pos__if__in__set,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_5061_card__length,axiom,
! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( set2 @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% card_length
thf(fact_5062_finite__lists__length__eq,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_5063_Re__complex__div__eq__0,axiom,
! [A2: complex,B2: complex] :
( ( ( re @ ( divide_divide @ complex @ A2 @ B2 ) )
= ( zero_zero @ real ) )
= ( ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) )
= ( zero_zero @ real ) ) ) ).
% Re_complex_div_eq_0
thf(fact_5064_Im__complex__div__eq__0,axiom,
! [A2: complex,B2: complex] :
( ( ( im @ ( divide_divide @ complex @ A2 @ B2 ) )
= ( zero_zero @ real ) )
= ( ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) )
= ( zero_zero @ real ) ) ) ).
% Im_complex_div_eq_0
thf(fact_5065_finite__lists__length__le,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_5066_Re__complex__div__lt__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less @ real @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_lt_0
thf(fact_5067_Re__complex__div__gt__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).
% Re_complex_div_gt_0
thf(fact_5068_Re__complex__div__ge__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).
% Re_complex_div_ge_0
thf(fact_5069_Re__complex__div__le__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq @ real @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_le_0
thf(fact_5070_Im__complex__div__lt__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less @ real @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_lt_0
thf(fact_5071_Im__complex__div__gt__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).
% Im_complex_div_gt_0
thf(fact_5072_Im__complex__div__ge__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).
% Im_complex_div_ge_0
thf(fact_5073_Im__complex__div__le__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq @ real @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_le_0
thf(fact_5074_card__lists__length__eq,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A3 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_5075_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_5076_complex__div__gt__0,axiom,
! [A2: complex,B2: complex] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) )
& ( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ) ).
% complex_div_gt_0
thf(fact_5077_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_5078_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_5079_VEBT__internal_Oset__n__deg__not__0,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,M: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ N ) ) ) ).
% VEBT_internal.set_n_deg_not_0
thf(fact_5080_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_5081_complex__div__cnj,axiom,
( ( divide_divide @ complex )
= ( ^ [A5: complex,B5: complex] : ( divide_divide @ complex @ ( times_times @ complex @ A5 @ ( 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_5082_card__lists__distinct__length__eq,axiom,
! [A: $tType,A3: set @ A,K: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A3 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_5083_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A3: set @ A] :
( ( ord_less @ nat @ K @ ( finite_card @ A @ A3 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_5084_sum__count__set,axiom,
! [A: $tType,Xs: list @ A,X5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X5 )
=> ( ( finite_finite2 @ A @ X5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X5 )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_5085_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
= ( distinct @ A @ Ys ) ) ).
% distinct_union
thf(fact_5086_count__notin,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_5087_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_5088_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ Xs ) ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_5089_finite__distinct__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [Xs3: list @ A] :
( ( ( set2 @ A @ Xs3 )
= A3 )
& ( distinct @ A @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_5090_distinct__card,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( finite_card @ A @ ( set2 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% distinct_card
thf(fact_5091_card__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( finite_card @ A @ ( set2 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Xs ) ) ).
% card_distinct
thf(fact_5092_count__le__length,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% count_le_length
thf(fact_5093_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_5094_num_Osize_I5_J,axiom,
! [X22: num] :
( ( size_size @ num @ ( bit0 @ X22 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(5)
thf(fact_5095_num_Osize_I6_J,axiom,
! [X32: num] :
( ( size_size @ num @ ( bit1 @ X32 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(6)
thf(fact_5096_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_5097_length__subseqs,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( subseqs @ A @ Xs ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_subseqs
thf(fact_5098_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% card_disjoint_shuffles
thf(fact_5099_finite__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] : ( finite_finite2 @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ).
% finite_shuffles
thf(fact_5100_subseqs__refl,axiom,
! [A: $tType,Xs: list @ A] : ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs_refl
thf(fact_5101_shuffles__commutes,axiom,
! [A: $tType] :
( ( shuffles @ A )
= ( ^ [Xs2: list @ A,Ys3: list @ A] : ( shuffles @ A @ Ys3 @ Xs2 ) ) ) ).
% shuffles_commutes
thf(fact_5102_subseqs__distinctD,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( ( distinct @ A @ Xs )
=> ( distinct @ A @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_5103_length__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( size_size @ ( list @ A ) @ Zs )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_5104_set__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( set2 @ A @ Zs )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_5105_distinct__set__subseqs,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ ( set @ A ) @ ( map @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_5106_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ A @ Ys )
=> ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( distinct @ A @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_5107_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_5108_horner__sum__bit__eq__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ).
% horner_sum_bit_eq_take_bit
thf(fact_5109_card__Pow,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( set @ A ) @ ( pow2 @ A @ A3 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A3 ) ) ) ) ).
% card_Pow
thf(fact_5110_Pow__empty,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_5111_Pow__singleton__iff,axiom,
! [A: $tType,X5: set @ A,Y6: set @ A] :
( ( ( pow2 @ A @ X5 )
= ( insert @ ( set @ A ) @ Y6 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
& ( Y6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_5112_Pow__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% Pow_iff
thf(fact_5113_PowI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B3 ) ) ) ).
% PowI
thf(fact_5114_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size @ ( list @ nat ) @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ I ) ) ).
% length_upt
thf(fact_5115_Pow__Int__eq,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( pow2 @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B3 ) ) ) ).
% Pow_Int_eq
thf(fact_5116_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F2: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F2 @ Xs ) @ N )
= ( F2 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_5117_map__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( map @ nat @ A @ ( nth @ A @ Xs ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= Xs ) ).
% map_nth
thf(fact_5118_Un__Pow__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B3 ) ) @ ( pow2 @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% Un_Pow_subset
thf(fact_5119_Pow__def,axiom,
! [A: $tType] :
( ( pow2 @ A )
= ( ^ [A4: set @ A] :
( collect @ ( set @ A )
@ ^ [B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% Pow_def
thf(fact_5120_Pow__top,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ A3 ) ) ).
% Pow_top
thf(fact_5121_PowD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% PowD
thf(fact_5122_Pow__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B3 ) ) ) ).
% Pow_mono
thf(fact_5123_Pow__bottom,axiom,
! [A: $tType,B3: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow2 @ A @ B3 ) ) ).
% Pow_bottom
thf(fact_5124_Pow__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( pow2 @ A @ A3 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_5125_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct @ nat @ ( upt @ I @ J ) ) ).
% distinct_upt
thf(fact_5126_nth__map__upt,axiom,
! [A: $tType,I: nat,N: nat,M: nat,F2: nat > A] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F2 @ ( upt @ M @ N ) ) @ I )
= ( F2 @ ( plus_plus @ nat @ M @ I ) ) ) ) ).
% nth_map_upt
thf(fact_5127_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z2: list @ A] : Y5 = Z2 )
= ( ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I4 )
= ( nth @ A @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_5128_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ? [X8: A] : ( P @ I4 @ X8 ) ) )
= ( ? [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ( P @ I4 @ ( nth @ A @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_5129_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_5130_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,M: nat,F2: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( minus_minus @ nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( F2 @ ( plus_plus @ nat @ M @ I2 ) ) ) )
=> ( ( map @ nat @ A @ F2 @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_5131_atLeastLessThan__upt,axiom,
( ( set_or7035219750837199246ssThan @ nat )
= ( ^ [I4: nat,J3: nat] : ( set2 @ nat @ ( upt @ I4 @ J3 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_5132_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map @ nat @ nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_5133_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_5134_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I2 ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_5135_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_5136_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_5137_nth__mem,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% nth_mem
thf(fact_5138_VEBT__internal_Ointhall,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,N: nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% VEBT_internal.inthall
thf(fact_5139_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I )
= ( nth @ A @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_5140_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs2: list @ A] :
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( I4 != J3 )
=> ( ( nth @ A @ Xs2 @ I4 )
!= ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_5141_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N3: nat,M3: nat] : ( set2 @ nat @ ( upt @ N3 @ ( suc @ M3 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_5142_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% atLeast_upt
thf(fact_5143_distinct__Ex1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [X3: nat] :
( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ X3 )
= X )
& ! [Y2: nat] :
( ( ( ord_less @ nat @ Y2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ Y2 )
= X ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_5144_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ I4 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M ) )
= ( upt @ N @ ( plus_plus @ nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_5145_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) ) ) ) ).
% atMost_upto
thf(fact_5146_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_5147_binomial__def,axiom,
( binomial
= ( ^ [N3: nat,K3: nat] :
( finite_card @ ( set @ nat )
@ ( collect @ ( set @ nat )
@ ^ [K7: set @ nat] :
( ( member @ ( set @ nat ) @ K7 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
& ( ( finite_card @ nat @ K7 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_5148_bij__betw__nth,axiom,
! [A: $tType,Xs: list @ A,A3: set @ nat,B3: set @ A] :
( ( distinct @ A @ Xs )
=> ( ( A3
= ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( ( B3
= ( set2 @ A @ Xs ) )
=> ( bij_betw @ nat @ A @ ( nth @ A @ Xs ) @ A3 @ B3 ) ) ) ) ).
% bij_betw_nth
thf(fact_5149_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A5: A,Xs2: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ ( nth @ B @ Xs2 @ N3 ) ) @ ( power_power @ A @ A5 @ N3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_5150_set__n__lists,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) )
= ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys3 )
= N )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_5151_set__update__distinct,axiom,
! [A: $tType,Xs: list @ A,N: nat,X: A] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_5152_list__update__overwrite,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A,Y: A] :
( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X ) @ I @ Y )
= ( list_update @ A @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_5153_length__list__update,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I @ X ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_list_update
thf(fact_5154_nth__list__update__neq,axiom,
! [A: $tType,I: nat,J: nat,Xs: list @ A,X: A] :
( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_5155_list__update__id,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_5156_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J )
=> ( ( nth @ nat @ ( upt @ I @ J ) @ K )
= ( plus_plus @ nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_5157_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( list_update @ A @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_5158_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_5159_set__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( set2 @ A @ Xs ) ) ) ) ).
% set_swap
thf(fact_5160_distinct__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( distinct @ A @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_5161_map__update,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,K: nat,Y: B] :
( ( map @ B @ A @ F2 @ ( list_update @ B @ Xs @ K @ Y ) )
= ( list_update @ A @ ( map @ B @ A @ F2 @ Xs ) @ K @ ( F2 @ Y ) ) ) ).
% map_update
thf(fact_5162_list__update__swap,axiom,
! [A: $tType,I: nat,I8: nat,Xs: list @ A,X: A,X7: A] :
( ( I != I8 )
=> ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X ) @ I8 @ X7 )
= ( list_update @ A @ ( list_update @ A @ Xs @ I8 @ X7 ) @ I @ X ) ) ) ).
% list_update_swap
thf(fact_5163_set__update__subsetI,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A,X: A,I: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_5164_set__update__subset__insert,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X ) ) @ ( insert @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_5165_set__update__memI,axiom,
! [A: $tType,N: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_5166_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_5167_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I @ X )
= Xs )
= ( ( nth @ A @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_5168_length__n__lists,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N @ Xs ) )
= ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_5169_length__n__lists__elem,axiom,
! [A: $tType,Ys: list @ A,N: nat,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_5170_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A2: A,I: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member @ A @ A2 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I @ A2 ) ) ) ) ).
% distinct_list_update
thf(fact_5171_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X5: set @ A,F2: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X5 )
=> ( ( finite_finite2 @ A @ X5 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( F2 @ X2 ) )
@ X5 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_5172_accp__subset,axiom,
! [A: $tType,R1: A > A > $o,R22: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R1 @ R22 )
=> ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R22 ) @ ( accp @ A @ R1 ) ) ) ).
% accp_subset
thf(fact_5173_accp__subset__induct,axiom,
! [A: $tType,D: A > $o,R2: A > A > $o,X: A,P: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ D @ ( accp @ A @ R2 ) )
=> ( ! [X3: A,Z: A] :
( ( D @ X3 )
=> ( ( R2 @ Z @ X3 )
=> ( D @ Z ) ) )
=> ( ( D @ X )
=> ( ! [X3: A] :
( ( D @ X3 )
=> ( ! [Z4: A] :
( ( R2 @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_5174_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y3: B] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_5175_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ns ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_5176_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_5177_sum__list__upt,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum_list_upt
thf(fact_5178_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_5179_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_5180_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% member_le_sum_list
thf(fact_5181_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C2: A,F2: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ Xs ) )
= ( times_times @ A @ C2 @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ) ).
% sum_list_const_mult
thf(fact_5182_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( ( semiring_0 @ A )
=> ! [F2: B > A,C2: A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ Xs ) )
= ( times_times @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) @ C2 ) ) ) ).
% sum_list_mult_const
thf(fact_5183_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: B > A,G: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ Xs ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum_list_addf
thf(fact_5184_sum__list__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,G: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ Xs ) )
= ( minus_minus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum_list_subtractf
thf(fact_5185_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_5186_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_5187_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_5188_sum__list__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [Xs: list @ A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( groups8242544230860333062m_list @ A @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ A @ A @ ( abs_abs @ A ) @ Xs ) ) ) ) ).
% sum_list_abs
thf(fact_5189_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F2: A > B,G: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ).
% sum_list_mono
thf(fact_5190_distinct__sum__list__conv__Sum,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( groups8242544230860333062m_list @ A @ Xs )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X2: A] : X2
@ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_5191_elem__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [K: nat,Ns: list @ A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Ns ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Ns @ K ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ).
% elem_le_sum_list
thf(fact_5192_sum__list__distinct__conv__sum__set,axiom,
! [C: $tType,B: $tType] :
( ( comm_monoid_add @ C )
=> ! [Xs: list @ B,F2: B > C] :
( ( distinct @ B @ Xs )
=> ( ( groups8242544230860333062m_list @ C @ ( map @ B @ C @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ B @ C @ F2 @ ( set2 @ B @ Xs ) ) ) ) ) ).
% sum_list_distinct_conv_sum_set
thf(fact_5193_sum_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ B,G: B > A] :
( ( distinct @ B @ Xs )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set2 @ B @ Xs ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum.distinct_set_conv_list
thf(fact_5194_sum__list__Suc,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X2: A] : ( suc @ ( F2 @ X2 ) )
@ Xs ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_5195_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R3: B,Xs: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X2: C] : R3
@ Xs ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R3 ) ) ) ).
% sum_list_triv
thf(fact_5196_sum__set__upt__conv__sum__list__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set2 @ nat @ ( upt @ M @ N ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ nat @ A @ F2 @ ( upt @ M @ N ) ) ) ) ) ).
% sum_set_upt_conv_sum_list_nat
thf(fact_5197_interv__sum__list__conv__sum__set__nat,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ! [F2: nat > B,M: nat,N: nat] :
( ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F2 @ ( upt @ M @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ B @ F2 @ ( set2 @ nat @ ( upt @ M @ N ) ) ) ) ) ).
% interv_sum_list_conv_sum_set_nat
thf(fact_5198_card__length__sum__list__rec,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N4 ) ) ) )
= ( plus_plus @ nat
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= ( minus_minus @ nat @ M @ ( one_one @ nat ) ) )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N4 ) ) ) )
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L2 ) @ ( one_one @ nat ) )
= N4 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_5199_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs2: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_5200_card__length__sum__list,axiom,
! [M: nat,N4: nat] :
( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N4 ) ) ) )
= ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N4 @ M ) @ ( one_one @ nat ) ) @ N4 ) ) ).
% card_length_sum_list
thf(fact_5201_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( F2 @ X2 ) )
@ ( set2 @ A @ Xs ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_5202_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% sum_list_update
thf(fact_5203_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
=> ( ( ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_5204_and__int_Opelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( 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 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( 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 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_5205_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F2: nat > B,Ns: list @ nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F2 @ Ns ) ) ) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
thf(fact_5206_sorted__wrt__map,axiom,
! [A: $tType,B: $tType,R2: A > A > $o,F2: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ R2 @ ( map @ B @ A @ F2 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X2: B,Y4: B] : ( R2 @ ( F2 @ X2 ) @ ( F2 @ Y4 ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_5207_strict__sorted__imp__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% strict_sorted_imp_sorted
thf(fact_5208_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R2 )
@ ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S2 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_5209_subrelI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S: 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 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S ) ) ).
% subrelI
thf(fact_5210_sorted__wrt__mono__rel,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o,Q: A > A > $o] :
( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted_wrt @ A @ P @ Xs )
=> ( sorted_wrt @ A @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_5211_strict__sorted__equal,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys )
=> ( ( ( set2 @ A @ Ys )
= ( set2 @ A @ Xs ) )
=> ( Ys = Xs ) ) ) ) ) ).
% strict_sorted_equal
thf(fact_5212_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R2 ) )
= ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_5213_sorted__wrt__true,axiom,
! [A: $tType,Xs: list @ A] :
( sorted_wrt @ A
@ ^ [Uu: A,Uv: A] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_5214_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R3: A,S: B,R2: set @ ( product_prod @ A @ B ),S7: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S ) @ R2 )
=> ( ( S7 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S7 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_5215_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R2 )
@ ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S2 ) )
= ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ) ) ).
% inf_Int_eq2
thf(fact_5216_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R2 )
@ ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S2 ) )
= ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ) ) ).
% sup_Un_eq2
thf(fact_5217_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_5218_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_5219_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_5220_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X2: B,Y4: B] : ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y4 ) )
@ Xs ) ) ) ).
% sorted_map
thf(fact_5221_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_5222_sorted__wrt01,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_5223_sorted__wrt__nth__less,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ P @ Xs )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_5224_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P3: A > A > $o,Xs2: list @ A] :
! [I4: nat,J3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P3 @ ( nth @ A @ Xs2 @ I4 ) @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_5225_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
=> ( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Xs )
= ( set2 @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_5226_sorted01,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted01
thf(fact_5227_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_5228_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [X3: list @ A] :
( ( ( set2 @ A @ X3 )
= A3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X3 )
& ( distinct @ A @ X3 )
& ! [Y2: list @ A] :
( ( ( ( set2 @ A @ Y2 )
= A3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y2 )
& ( distinct @ A @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_5229_sorted__wrt__less__idx,axiom,
! [Ns: list @ nat,I: nat] :
( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ nat ) @ Ns ) )
=> ( ord_less_eq @ nat @ I @ ( nth @ nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_5230_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ ( suc @ I4 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_5231_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_5232_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_5233_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ~ ! [L3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L3 )
=> ( ( ( set2 @ A @ L3 )
= A3 )
=> ( ( size_size @ ( list @ A ) @ L3 )
!= ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_5234_and__int_Opinduct,axiom,
! [A0: int,A12: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) )
=> ( ! [K2: int,L3: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L3 ) )
=> ( ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K2 @ L3 ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% and_int.pinduct
thf(fact_5235_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R3: A,Q3: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R3 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R3 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ ( 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 @ Q3 @ R3 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ R3 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_5236_Divides_Oadjust__div__eq,axiom,
! [Q3: int,R3: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q3 @ R3 ) )
= ( plus_plus @ int @ Q3
@ ( zero_neq_one_of_bool @ int
@ ( R3
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_5237_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q3: int,R3: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A2 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q3 @ R3 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_5238_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
~ ( bot_bot @ ( A > B > $o ) @ X @ Y ) ).
% bot2E
thf(fact_5239_less__by__empty,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] :
( ( A3
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B3 ) ) ).
% less_by_empty
thf(fact_5240_xor__num_Ocases,axiom,
! [X: product_prod @ num @ num] :
( ( X
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ one2 ) )
=> ( ! [M4: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) )
=> ( ! [M4: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ one2 ) )
=> ( ! [M4: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) )
=> ~ ! [M4: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_5241_eucl__rel__int__by0,axiom,
! [K: int] : ( eucl_rel_int @ K @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K ) ) ).
% eucl_rel_int_by0
thf(fact_5242_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q3: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K
= ( times_times @ int @ Q3 @ L ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_5243_zminus1__lemma,axiom,
! [A2: int,B2: int,Q3: int,R3: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R3 ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A2 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R3
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q3 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q3 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R3 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_5244_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q3: int,R3: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R3 ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L @ Q3 ) @ 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 ) )
=> ( Q3
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_5245_eucl__rel__int__remainderI,axiom,
! [R3: int,L: int,K: int,Q3: int] :
( ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ L ) )
=> ( ( K
= ( plus_plus @ int @ ( times_times @ int @ Q3 @ L ) @ R3 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R3 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_5246_eucl__rel__int_Ocases,axiom,
! [A12: int,A23: int,A32: product_prod @ int @ int] :
( ( eucl_rel_int @ A12 @ A23 @ A32 )
=> ( ( ( A23
= ( zero_zero @ int ) )
=> ( A32
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A12 ) ) )
=> ( ! [Q5: int] :
( ( A32
= ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
=> ( ( A23
!= ( zero_zero @ int ) )
=> ( A12
!= ( times_times @ int @ Q5 @ A23 ) ) ) )
=> ~ ! [R: int,Q5: int] :
( ( A32
= ( product_Pair @ int @ int @ Q5 @ R ) )
=> ( ( ( sgn_sgn @ int @ R )
= ( sgn_sgn @ int @ A23 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ A23 ) )
=> ( A12
!= ( plus_plus @ int @ ( times_times @ int @ Q5 @ A23 ) @ R ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_5247_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A13: int,A24: int,A33: product_prod @ int @ int] :
( ? [K3: int] :
( ( A13 = K3 )
& ( A24
= ( zero_zero @ int ) )
& ( A33
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L2: int,K3: int,Q6: int] :
( ( A13 = K3 )
& ( A24 = L2 )
& ( A33
= ( product_Pair @ int @ int @ Q6 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q6 @ L2 ) ) )
| ? [R5: int,L2: int,K3: int,Q6: int] :
( ( A13 = K3 )
& ( A24 = L2 )
& ( A33
= ( product_Pair @ int @ int @ Q6 @ R5 ) )
& ( ( sgn_sgn @ int @ R5 )
= ( sgn_sgn @ int @ L2 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L2 ) )
& ( K3
= ( plus_plus @ int @ ( times_times @ int @ Q6 @ L2 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_5248_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q3: int,R3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R3 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R3 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_5249_upto_Opinduct,axiom,
! [A0: int,A12: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) )
=> ( ! [I2: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I2 @ J2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J2 ) )
=> ( P @ I2 @ J2 ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% upto.pinduct
thf(fact_5250_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( ( ord_less @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_5251_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( ( ord_less_eq @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_5252_minus__numeral__div__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% minus_numeral_div_numeral
thf(fact_5253_numeral__div__minus__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% numeral_div_minus_numeral
thf(fact_5254_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num] :
( ( unique8689654367752047608divmod @ A @ M @ one2 )
= ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M ) @ ( zero_zero @ A ) ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_5255_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_5256_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_5257_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_5258_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_5259_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M3: num,N3: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M3 ) @ ( numeral_numeral @ nat @ N3 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M3 ) @ ( numeral_numeral @ nat @ N3 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_5260_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M3: num,N3: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M3 ) @ ( numeral_numeral @ int @ N3 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M3 ) @ ( numeral_numeral @ int @ N3 ) ) ) ) ) ).
% divmod_int_def
thf(fact_5261_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M3: num,N3: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M3 ) @ ( numeral_numeral @ A @ N3 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M3 ) @ ( numeral_numeral @ A @ N3 ) ) ) ) ) ) ).
% divmod_def
thf(fact_5262_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M3: num,N3: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M3 @ N3 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M3 ) ) @ ( unique1321980374590559556d_step @ A @ N3 @ ( unique8689654367752047608divmod @ A @ M3 @ ( bit0 @ N3 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_5263_dvd__numeral__simp,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( unique5940410009612947441es_aux @ A @ ( unique8689654367752047608divmod @ A @ N @ M ) ) ) ) ).
% dvd_numeral_simp
thf(fact_5264_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: A,R3: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q3 @ R3 ) )
= ( R3
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_5265_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_5266_length__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys ) )
= ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_product
thf(fact_5267_distinct__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ B @ Ys )
=> ( distinct @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_5268_in__measure,axiom,
! [A: $tType,X: A,Y: A,F2: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measure @ A @ F2 ) )
= ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).
% in_measure
thf(fact_5269_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ 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 ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ 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 ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_5270_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F2: nat > A > A,A2: nat,B2: nat,Acc2: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A2 @ ( product_Pair @ nat @ A @ B2 @ Acc2 ) ) ) )
=> ( ( ( ord_less @ nat @ B2 @ A2 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A2 @ B2 @ Acc2 )
= Acc2 ) )
& ( ~ ( ord_less @ nat @ B2 @ A2 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A2 @ B2 @ Acc2 )
= ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A2 @ ( one_one @ nat ) ) @ B2 @ ( F2 @ A2 @ Acc2 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_5271_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A12: nat,A23: nat,A32: A,P: ( nat > A > A ) > nat > nat > A > $o] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A12 @ ( product_Pair @ nat @ A @ A23 @ A32 ) ) ) )
=> ( ! [F5: nat > A > A,A6: nat,B6: 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 ) ) @ F5 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A6 @ ( product_Pair @ nat @ A @ B6 @ Acc3 ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B6 @ A6 )
=> ( P @ F5 @ ( plus_plus @ nat @ A6 @ ( one_one @ nat ) ) @ B6 @ ( F5 @ A6 @ Acc3 ) ) )
=> ( P @ F5 @ A6 @ B6 @ Acc3 ) ) )
=> ( P @ A0 @ A12 @ A23 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_5272_in__finite__psubset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A3 @ B3 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
& ( finite_finite2 @ A @ B3 ) ) ) ).
% in_finite_psubset
thf(fact_5273_divmod__BitM__2__eq,axiom,
! [M: num] :
( ( unique8689654367752047608divmod @ int @ ( bitM @ M ) @ ( bit0 @ one2 ) )
= ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% divmod_BitM_2_eq
thf(fact_5274_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R5: A] : ( product_Pair @ A @ A @ Q6 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_5275_dbl__dec__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% dbl_dec_simps(5)
thf(fact_5276_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_5277_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R5: A] : ( product_Pair @ A @ A @ Q6 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_5278_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q3: product_prod @ A @ B,F2: A > B > C,G: A > B > C,P5: product_prod @ A @ B] :
( ! [X3: A,Y3: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y3 )
= Q3 )
=> ( ( F2 @ X3 @ Y3 )
= ( G @ X3 @ Y3 ) ) )
=> ( ( P5 = Q3 )
=> ( ( product_case_prod @ A @ B @ C @ F2 @ P5 )
= ( product_case_prod @ A @ B @ C @ G @ Q3 ) ) ) ) ).
% split_cong
thf(fact_5279_semiring__norm_I26_J,axiom,
( ( bitM @ one2 )
= one2 ) ).
% semiring_norm(26)
thf(fact_5280_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_5281_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_5282_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_5283_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_5284_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_5285_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_5286_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_5287_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_5288_odd__numeral__BitM,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [W2: num] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bitM @ W2 ) ) ) ) ).
% odd_numeral_BitM
thf(fact_5289_not__numeral__BitM__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bitM @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_5290_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q6: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_5291_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q6: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_5292_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_5293_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A2: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F2 @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_5294_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M3: nat,N3: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N3
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M3 @ N3 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M3 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q6: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q6 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_5295_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ~ ! [L3: list @ B] :
( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ L3 ) )
=> ( ( ( set2 @ B @ L3 )
= A3 )
=> ( ( size_size @ ( list @ B ) @ L3 )
!= ( finite_card @ B @ A3 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_5296_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P5: product_prod @ A @ B,Z3: C,C2: A > B > ( set @ C )] :
( ! [A6: A,B6: B] :
( ( P5
= ( product_Pair @ A @ B @ A6 @ B6 ) )
=> ( member @ C @ Z3 @ ( C2 @ A6 @ B6 ) ) )
=> ( member @ C @ Z3 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P5 ) ) ) ).
% mem_case_prodI2
thf(fact_5297_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z3: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z3 @ ( C2 @ A2 @ B2 ) )
=> ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_5298_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P5: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
( ! [A6: A,B6: B] :
( ( ( product_Pair @ A @ B @ A6 @ B6 )
= P5 )
=> ( C2 @ A6 @ B6 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ X ) ) ).
% case_prodI2'
thf(fact_5299_case__prodI2,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B,C2: A > B > $o] :
( ! [A6: A,B6: B] :
( ( P5
= ( product_Pair @ A @ B @ A6 @ B6 ) )
=> ( C2 @ A6 @ B6 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P5 ) ) ).
% case_prodI2
thf(fact_5300_case__prodI,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A2: A,B2: B] :
( ( F2 @ A2 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_5301_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,Y7: B] :
( ( X = X9 )
& ( Y = Y7 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% The_split_eq
thf(fact_5302_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,Y7: B] :
( ( X = X9 )
& ( Y = Y7 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% Eps_case_prod_eq
thf(fact_5303_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P5: product_prod @ A @ B,Z3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ Z3 )
=> ~ ! [X3: A,Y3: B] :
( ( P5
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 @ Z3 ) ) ) ).
% case_prodE'
thf(fact_5304_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R2: A > B > C > $o,A2: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R2 @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
=> ( R2 @ A2 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_5305_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P5: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P5 )
=> ~ ! [X3: A,Y3: B] :
( ( P5
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_5306_case__prodD,axiom,
! [A: $tType,B: $tType,F2: A > B > $o,A2: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( F2 @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_5307_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P3: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B5: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B5 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_5308_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A3: A > B > $o,B3: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B3 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_5309_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_5310_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_5311_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_5312_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_5313_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) )
=> ( distinct @ B @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_5314_finite__psubset__def,axiom,
! [A: $tType] :
( ( finite_psubset @ A )
= ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A4: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
& ( finite_finite2 @ A @ B4 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_5315_prod_Ocase__distrib,axiom,
! [C: $tType,D3: $tType,B: $tType,A: $tType,H2: C > D3,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( H2 @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( product_case_prod @ A @ B @ D3
@ ^ [X12: A,X24: B] : ( H2 @ ( F2 @ X12 @ X24 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_5316_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q6: int,R5: int] :
( plus_plus @ int @ Q6
@ ( zero_neq_one_of_bool @ int
@ ( R5
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_5317_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z3: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z3 ) )
=> ~ ! [X3: B,Y3: C] :
( ( Z3
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_5318_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X2: A,Y4: B] : ( F2 @ ( product_Pair @ A @ B @ X2 @ Y4 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_5319_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y3: B] :
( ( F2 @ X3 @ Y3 )
= ( G @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F2 )
= G ) ) ).
% cond_case_prod_eta
thf(fact_5320_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N3: nat] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M3: nat,Q6: nat] :
( if @ A
@ ( Q6
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M3 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M3 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_5321_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A3: set @ B,L: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ L ) )
& ( ( set2 @ B @ L )
= A3 )
& ( ( size_size @ ( list @ B ) @ L )
= ( finite_card @ B @ A3 ) ) )
= ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A3 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_5322_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z8: int,Z7: int] :
( ( ord_less_eq @ int @ D5 @ Z7 )
& ( ord_less @ int @ Z8 @ Z7 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_5323_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B5: B] :
( P
& ( Q @ A5 @ B5 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_5324_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_5325_linorder_Osorted__key__list__of__set_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( sorted8670434370408473282of_set @ A @ B )
= ( sorted8670434370408473282of_set @ A @ B ) ) ).
% linorder.sorted_key_list_of_set.cong
thf(fact_5326_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A3: set @ B,B3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ S2 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A3 )
= ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ B3 ) )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( A3 = B3 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_inject
thf(fact_5327_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( set2 @ B @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A3 ) )
= A3 ) ) ) ) ).
% folding_insort_key.set_sorted_key_list_of_set
thf(fact_5328_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S2 )
=> ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A3 ) )
= ( finite_card @ B @ A3 ) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
thf(fact_5329_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S2 )
=> ( distinct @ A @ ( map @ B @ A @ F2 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A3 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_5330_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S2 )
=> ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F2 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A3 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_5331_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S2 )
=> ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A3 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_5332_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( set2 @ B @ Xs ) @ S2 )
=> ( ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F2 @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( set2 @ B @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_5333_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z8: int,Z7: int] :
( ( ord_less_eq @ int @ D5 @ Z8 )
& ( ord_less @ int @ Z8 @ Z7 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_5334_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,X: B,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A3 ) @ S2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( insert @ B @ X @ A3 ) )
= ( insort_key @ A @ B @ Less_eq @ F2 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_5335_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,X: B,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A3 ) @ S2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( insert @ B @ X @ A3 ) )
= ( insort_key @ A @ B @ Less_eq @ F2 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A3 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
thf(fact_5336_linorder_Oinsort__key_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( insort_key @ A @ B )
= ( insort_key @ A @ B ) ) ).
% linorder.insort_key.cong
thf(fact_5337_rat__inverse__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( A5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A5 ) @ B5 ) @ ( abs_abs @ int @ A5 ) ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_inverse_code
thf(fact_5338_case__prod__app,axiom,
! [A: $tType,D3: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D3 > A ) )
= ( ^ [F3: B > C > D3 > A,X2: product_prod @ B @ C,Y4: D3] :
( product_case_prod @ B @ C @ A
@ ^ [L2: B,R5: C] : ( F3 @ L2 @ R5 @ Y4 )
@ X2 ) ) ) ).
% case_prod_app
thf(fact_5339_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,X: B,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A3 ) @ S2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A3 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_5340_in__set__remove1,axiom,
! [A: $tType,A2: A,B2: A,Xs: list @ A] :
( ( A2 != B2 )
=> ( ( member @ A @ A2 @ ( set2 @ A @ ( remove1 @ A @ B2 @ Xs ) ) )
= ( member @ A @ A2 @ ( set2 @ A @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_5341_quotient__of__number_I3_J,axiom,
! [K: num] :
( ( quotient_of @ ( numeral_numeral @ rat @ K ) )
= ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(3)
thf(fact_5342_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_5343_set__remove1__eq,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( set2 @ A @ ( remove1 @ A @ X @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_remove1_eq
thf(fact_5344_quotient__of__number_I5_J,axiom,
! [K: num] :
( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(5)
thf(fact_5345_distinct__remove1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( remove1 @ A @ X @ Xs ) ) ) ).
% distinct_remove1
thf(fact_5346_remove1__commute,axiom,
! [A: $tType,X: A,Y: A,Zs: list @ A] :
( ( remove1 @ A @ X @ ( remove1 @ A @ Y @ Zs ) )
= ( remove1 @ A @ Y @ ( remove1 @ A @ X @ Zs ) ) ) ).
% remove1_commute
thf(fact_5347_remove1__idem,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_5348_notin__set__remove1,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member @ A @ X @ ( set2 @ A @ ( remove1 @ A @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_5349_set__remove1__subset,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( remove1 @ A @ X @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_remove1_subset
thf(fact_5350_sorted__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A2 @ Xs ) ) ) ) ).
% sorted_remove1
thf(fact_5351_quotient__of__denom__pos,axiom,
! [R3: rat,P5: int,Q3: int] :
( ( ( quotient_of @ R3 )
= ( product_Pair @ int @ int @ P5 @ Q3 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% quotient_of_denom_pos
thf(fact_5352_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_5353_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R2: $o,X: A,Y: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R2 )
=> ( R2
& ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ) ).
% predicate2D_conj
thf(fact_5354_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F2: ( A > B ) > C,G: C] :
( ( F2
= ( ^ [X2: A > B] : G ) )
=> ( ( F2
@ ^ [X2: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_5355_length__remove1,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_remove1
thf(fact_5356_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [X: B,Xs: list @ B,F2: B > A] :
( ( member @ B @ X @ ( set2 @ B @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) )
= ( plus_plus @ A @ ( F2 @ X ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_5357_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ ^ [A5: A,B5: A] :
( ( P @ A5 @ B5 )
| ( A5 = B5 ) ) ) ).
% eq_subset
thf(fact_5358_rat__uminus__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A5 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_uminus_code
thf(fact_5359_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P6: rat,Q6: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B5: int,D5: int] : ( ord_less @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C6 @ B5 ) )
@ ( quotient_of @ Q6 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_code
thf(fact_5360_rat__floor__code,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [P6: rat] : ( product_case_prod @ int @ int @ int @ ( divide_divide @ int ) @ ( quotient_of @ P6 ) ) ) ) ).
% rat_floor_code
thf(fact_5361_rat__abs__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A5 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_abs_code
thf(fact_5362_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P6: rat,Q6: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B5: int,D5: int] : ( ord_less_eq @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C6 @ B5 ) )
@ ( quotient_of @ Q6 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_eq_code
thf(fact_5363_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P5 )
= P5 ) ).
% case_prod_Pair_iden
thf(fact_5364_rat__plus__code,axiom,
! [P5: rat,Q3: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P5 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ B5 @ C6 ) ) @ ( times_times @ int @ C6 @ D5 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_plus_code
thf(fact_5365_rat__minus__code,axiom,
! [P5: rat,Q3: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P5 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ B5 @ C6 ) ) @ ( times_times @ int @ C6 @ D5 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_minus_code
thf(fact_5366_VEBT__internal_Ofoldr__zero,axiom,
! [Xs: list @ nat,D4: nat] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ nat ) @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nth @ nat @ Xs @ I2 ) ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ nat ) @ Xs ) @ ( minus_minus @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs @ D4 ) @ D4 ) ) ) ).
% VEBT_internal.foldr_zero
thf(fact_5367_normalize__denom__zero,axiom,
! [P5: int] :
( ( normalize @ ( product_Pair @ int @ int @ P5 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_5368_normalize__negative,axiom,
! [Q3: int,P5: int] :
( ( ord_less @ int @ Q3 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q3 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P5 ) @ ( uminus_uminus @ int @ Q3 ) ) ) ) ) ).
% normalize_negative
thf(fact_5369_foldr__cong,axiom,
! [B: $tType,A: $tType,A2: A,B2: A,L: list @ B,K: list @ B,F2: B > A > A,G: B > A > A] :
( ( A2 = B2 )
=> ( ( L = K )
=> ( ! [A6: A,X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ L ) )
=> ( ( F2 @ X3 @ A6 )
= ( G @ X3 @ A6 ) ) )
=> ( ( foldr @ B @ A @ F2 @ L @ A2 )
= ( foldr @ B @ A @ G @ K @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_5370_VEBT__internal_Ofoldr__one,axiom,
! [D4: nat,Ys: list @ nat] : ( ord_less_eq @ nat @ D4 @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Ys @ D4 ) ) ).
% VEBT_internal.foldr_one
thf(fact_5371_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs2: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs2 @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_5372_normalize__denom__pos,axiom,
! [R3: product_prod @ int @ int,P5: int,Q3: int] :
( ( ( normalize @ R3 )
= ( product_Pair @ int @ int @ P5 @ Q3 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% normalize_denom_pos
thf(fact_5373_normalize__crossproduct,axiom,
! [Q3: int,S: int,P5: int,R3: int] :
( ( Q3
!= ( zero_zero @ int ) )
=> ( ( S
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q3 ) )
= ( normalize @ ( product_Pair @ int @ int @ R3 @ S ) ) )
=> ( ( times_times @ int @ P5 @ S )
= ( times_times @ int @ R3 @ Q3 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_5374_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A5: A,Xs2: list @ B] :
( foldr @ B @ A
@ ^ [X2: B,B5: A] : ( plus_plus @ A @ ( F3 @ X2 ) @ ( times_times @ A @ A5 @ B5 ) )
@ Xs2
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_5375_VEBT__internal_Ofoldr__same__int,axiom,
! [Xs: list @ nat,Y: nat] :
( ! [X3: nat,Y3: nat] :
( ( member @ nat @ X3 @ ( set2 @ nat @ Xs ) )
=> ( ( member @ nat @ Y3 @ ( set2 @ nat @ Xs ) )
=> ( X3 = Y3 ) ) )
=> ( ! [X3: nat] :
( ( member @ nat @ X3 @ ( set2 @ nat @ Xs ) )
=> ( X3 = Y ) )
=> ( ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs @ ( zero_zero @ nat ) )
= ( times_times @ nat @ ( size_size @ ( list @ nat ) @ Xs ) @ Y ) ) ) ) ).
% VEBT_internal.foldr_same_int
thf(fact_5376_VEBT__internal_Ofoldr__mono,axiom,
! [Xs: list @ nat,Ys: list @ nat,C2: nat,D4: nat] :
( ( ( size_size @ ( list @ nat ) @ Xs )
= ( size_size @ ( list @ nat ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ nat ) @ Xs ) )
=> ( ord_less @ nat @ ( nth @ nat @ Xs @ I2 ) @ ( nth @ nat @ Ys @ I2 ) ) )
=> ( ( ord_less_eq @ nat @ C2 @ D4 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Xs @ C2 ) @ ( size_size @ ( list @ nat ) @ Ys ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ Ys @ D4 ) ) ) ) ) ).
% VEBT_internal.foldr_mono
thf(fact_5377_VEBT__internal_Olist__every__elemnt__bound__sum__bound,axiom,
! [A: $tType,Xs: list @ A,F2: A > nat,Bound: nat,I: nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F2 @ X3 ) @ Bound ) )
=> ( ord_less_eq @ nat @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ A @ nat @ F2 @ Xs ) @ I ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ Bound ) @ I ) ) ) ).
% VEBT_internal.list_every_elemnt_bound_sum_bound
thf(fact_5378_rat__times__code,axiom,
! [P5: rat,Q3: rat] :
( ( quotient_of @ ( times_times @ rat @ P5 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ B5 ) @ ( times_times @ int @ C6 @ D5 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_times_code
thf(fact_5379_rat__divide__code,axiom,
! [P5: rat,Q3: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P5 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C6 @ B5 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_divide_code
thf(fact_5380_VEBT__internal_Ospace_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_space @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_space @ Summary ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space @ TreeList ) @ ( zero_zero @ nat ) ) ) ) ).
% VEBT_internal.space.simps(2)
thf(fact_5381_Frct__code__post_I5_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K ) ) )
= ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K ) ) ) ).
% Frct_code_post(5)
thf(fact_5382_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_space2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_VEBT_space2 @ Summary ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space2 @ TreeList ) @ ( zero_zero @ nat ) ) ) ) ).
% VEBT_internal.space'.simps(2)
thf(fact_5383_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_cnt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
= ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( vEBT_VEBT_cnt @ Summary ) ) @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ vEBT_VEBT @ real @ vEBT_VEBT_cnt @ TreeList ) @ ( zero_zero @ real ) ) ) ) ).
% VEBT_internal.cnt.simps(2)
thf(fact_5384_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_5385_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_5386_VEBT__internal_Oheight__compose__summary,axiom,
! [Summary: vEBT_VEBT,Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ Summary ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).
% VEBT_internal.height_compose_summary
thf(fact_5387_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_5388_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_5389_VEBT__internal_Olistsum__bound,axiom,
! [A: $tType,Xs: list @ A,F2: A > real,Y: real] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ real @ Y @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F2 @ Xs ) @ Y ) ) ) ).
% VEBT_internal.listsum_bound
thf(fact_5390_VEBT__internal_Ofoldr__same,axiom,
! [Xs: list @ real,Y: real] :
( ! [X3: real,Y3: real] :
( ( member @ real @ X3 @ ( set2 @ real @ Xs ) )
=> ( ( member @ real @ Y3 @ ( set2 @ real @ Xs ) )
=> ( X3 = Y3 ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set2 @ real @ Xs ) )
=> ( X3 = Y ) )
=> ( ( foldr @ real @ real @ ( plus_plus @ real ) @ Xs @ ( zero_zero @ real ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( list @ real ) @ Xs ) ) @ Y ) ) ) ) ).
% VEBT_internal.foldr_same
thf(fact_5391_VEBT__internal_Oheight__compose__child,axiom,
! [T2: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Info: option @ ( product_prod @ nat @ nat ),Deg: nat,Summary: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ T2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ) ).
% VEBT_internal.height_compose_child
thf(fact_5392_Frct__code__post_I2_J,axiom,
! [A2: int] :
( ( frct @ ( product_Pair @ int @ int @ A2 @ ( zero_zero @ int ) ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(2)
thf(fact_5393_Frct__code__post_I1_J,axiom,
! [A2: int] :
( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A2 ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(1)
thf(fact_5394_Frct__code__post_I6_J,axiom,
! [K: num,L: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L ) ) )
= ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_5395_VEBT__internal_Of__g__map__foldr__bound,axiom,
! [A: $tType,Xs: list @ A,F2: A > real,C2: real,G: A > real,D4: real] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ real @ ( F2 @ X3 ) @ ( times_times @ real @ C2 @ ( G @ X3 ) ) ) )
=> ( ord_less_eq @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F2 @ Xs ) @ D4 ) @ ( plus_plus @ real @ ( times_times @ real @ C2 @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ G @ Xs ) @ ( zero_zero @ real ) ) ) @ D4 ) ) ) ).
% VEBT_internal.f_g_map_foldr_bound
thf(fact_5396_VEBT__internal_Oreal__nat__list,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A,C2: nat] :
( ( semiring_1_of_nat @ real @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ A @ nat @ F2 @ Xs ) @ C2 ) )
= ( foldr @ real @ real @ ( plus_plus @ real )
@ ( map @ A @ real
@ ^ [X2: A] : ( semiring_1_of_nat @ real @ ( F2 @ X2 ) )
@ Xs )
@ ( semiring_1_of_nat @ real @ C2 ) ) ) ).
% VEBT_internal.real_nat_list
thf(fact_5397_VEBT__internal_Olist__every__elemnt__bound__sum__bound__real,axiom,
! [A: $tType,Xs: list @ A,F2: A > real,Bound: real,I: real] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ real @ ( F2 @ X3 ) @ Bound ) )
=> ( ord_less_eq @ real @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ A @ real @ F2 @ Xs ) @ I ) @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( list @ A ) @ Xs ) ) @ Bound ) @ I ) ) ) ).
% VEBT_internal.list_every_elemnt_bound_sum_bound_real
thf(fact_5398_Frct__code__post_I4_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) )
= ( numeral_numeral @ rat @ K ) ) ).
% Frct_code_post(4)
thf(fact_5399_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
! [Deg: nat,X: nat,Info: option @ ( product_prod @ nat @ nat ),TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) @ X )
=> ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) )
& ( ~ ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) @ X )
=> ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% VEBT_internal.insert'.simps(2)
thf(fact_5400_vebt__insert_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) @ X )
= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) ) ).
% vebt_insert.simps(3)
thf(fact_5401_VEBT__internal_Ospace_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space @ X )
= Y )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space @ TreeList2 ) @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.space.elims
thf(fact_5402_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_cnt2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_cnt2 @ Summary ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_cnt2 @ TreeList ) @ ( zero_zero @ nat ) ) ) ) ).
% VEBT_internal.cnt'.simps(2)
thf(fact_5403_VEBT_Osize_I4_J,axiom,
! [X21: $o,X222: $o] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size(4)
thf(fact_5404_VEBT__internal_OLeaf__0__not,axiom,
! [A2: $o,B2: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) ) ).
% VEBT_internal.Leaf_0_not
thf(fact_5405_VEBT__internal_Oheight_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( zero_zero @ nat ) ) ).
% VEBT_internal.height.simps(1)
thf(fact_5406_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_5407_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B6: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ 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 ),V4: nat,TreeList2: list @ vEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S3 ) @ X3 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_5408_invar__vebt_Ointros_I1_J,axiom,
! [A2: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ).
% invar_vebt.intros(1)
thf(fact_5409_vebt__delete__code_I1_J,axiom,
! [X: nat,A2: $o,B2: $o] :
( ( ( X
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ $false @ B2 ) ) )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ A2 @ $false ) ) )
& ( ( X
!= ( one_one @ nat ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ) ) ).
% vebt_delete_code(1)
thf(fact_5410_vebt__insert__code_I1_J,axiom,
! [X: nat,A2: $o,B2: $o] :
( ( ( X
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ $true @ B2 ) ) )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ A2 @ $true ) ) )
& ( ( X
!= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ) ) ).
% vebt_insert_code(1)
thf(fact_5411_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_5412_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
thf(fact_5413_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
thf(fact_5414_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
! [Uu3: $o,Uv3: $o] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ Uu3 @ Uv3 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
thf(fact_5415_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
! [Uu3: $o,Uv3: $o] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ Uu3 @ Uv3 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
thf(fact_5416_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
! [Uu3: $o,B2: $o] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uu3 @ B2 ) @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
thf(fact_5417_VEBT__internal_Ocnt_H_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_cnt2 @ X )
= Y )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_cnt2 @ TreeList2 ) @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.cnt'.elims
thf(fact_5418_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
thf(fact_5419_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
thf(fact_5420_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_VEBT_space2 @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ).
% VEBT_internal.space'.simps(1)
thf(fact_5421_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
! [A2: $o,Uw2: $o] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A2 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
thf(fact_5422_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
! [Uu3: $o,B2: $o] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uu3 @ B2 ) @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
thf(fact_5423_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( X
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
thf(fact_5424_VEBT__internal_Ospace_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_VEBT_space @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% VEBT_internal.space.simps(1)
thf(fact_5425_VEBT__internal_Ocnt__cnt__eq,axiom,
( vEBT_VEBT_cnt
= ( ^ [T3: vEBT_VEBT] : ( semiring_1_of_nat @ real @ ( vEBT_VEBT_cnt2 @ T3 ) ) ) ) ).
% VEBT_internal.cnt_cnt_eq
thf(fact_5426_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
! [A2: $o,Uw2: $o] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A2 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
thf(fact_5427_VEBT__internal_Ocnt_Oelims,axiom,
! [X: vEBT_VEBT,Y: real] :
( ( ( vEBT_VEBT_cnt @ X )
= Y )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( Y
!= ( one_one @ real ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ vEBT_VEBT @ real @ vEBT_VEBT_cnt @ TreeList2 ) @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% VEBT_internal.cnt.elims
thf(fact_5428_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( X
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
thf(fact_5429_VEBT__internal_Oinsert_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_insert @ X @ Xa )
= Y )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( Y
!= ( vEBT_vebt_insert @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa )
=> ( Y
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa )
=> ( Y
= ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% VEBT_internal.insert'.elims
thf(fact_5430_VEBT__internal_Ospace_H_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space2 @ X )
= Y )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space2 @ TreeList2 ) @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.space'.elims
thf(fact_5431_vebt__insert_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) @ X )
= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) ) ).
% vebt_insert.simps(2)
thf(fact_5432_VEBT__internal_Ospace_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ X )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A6 @ B6 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space @ TreeList2 ) @ ( zero_zero @ nat ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.space.pelims
thf(fact_5433_VEBT__internal_Ospace_H_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_space2 @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ X )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A6 @ B6 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_space2 @ TreeList2 ) @ ( zero_zero @ nat ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.space'.pelims
thf(fact_5434_VEBT__internal_Oinsert_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_VEBT_insert @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Y
= ( vEBT_vebt_insert @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa )
=> ( Y
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) @ Xa )
=> ( Y
= ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% VEBT_internal.insert'.pelims
thf(fact_5435_VEBT__internal_Ocnt_Opelims,axiom,
! [X: vEBT_VEBT,Y: real] :
( ( ( vEBT_VEBT_cnt @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Y
= ( one_one @ real ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Leaf @ A6 @ B6 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr @ real @ real @ ( plus_plus @ real ) @ ( map @ vEBT_VEBT @ real @ vEBT_VEBT_cnt @ TreeList2 ) @ ( zero_zero @ real ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.cnt.pelims
thf(fact_5436_VEBT__internal_Ocnt_H_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_cnt2 @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ X )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A6 @ B6 ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr @ nat @ nat @ ( plus_plus @ nat ) @ ( map @ vEBT_VEBT @ nat @ vEBT_VEBT_cnt2 @ TreeList2 ) @ ( zero_zero @ nat ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.cnt'.pelims
thf(fact_5437_vebt__delete_Osimps_I2_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ A2 @ $false ) ) ).
% vebt_delete.simps(2)
thf(fact_5438_vebt__delete_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ B2 ) ) ).
% vebt_delete.simps(1)
thf(fact_5439_VEBT__internal_Omi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
& ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) ) ) ).
% VEBT_internal.mi_ma_2_deg
thf(fact_5440_int__of__nat__def,axiom,
( code_T6385005292777649522of_nat
= ( semiring_1_of_nat @ int ) ) ).
% int_of_nat_def
thf(fact_5441_VEBT_Osize_I3_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X122: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Node @ X11 @ X122 @ 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_5442_size__list__estimation,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F2: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less @ nat @ Y @ ( F2 @ X ) )
=> ( ord_less @ nat @ Y @ ( size_list @ A @ F2 @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_5443_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F2: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ Y @ ( F2 @ X ) )
=> ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F2 @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_5444_size__list__pointwise,axiom,
! [A: $tType,Xs: list @ A,F2: A > nat,G: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F2 @ Xs ) @ ( size_list @ A @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_5445_vebt__delete_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TrLst: list @ vEBT_VEBT,Smry: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TrLst @ Smry ) ) ).
% vebt_delete.simps(5)
thf(fact_5446_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
! [V2: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Vf )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
thf(fact_5447_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
! [V2: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vc @ Vd ) @ Ve )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
thf(fact_5448_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
! [V2: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Vf )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
thf(fact_5449_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
! [V2: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vc @ Vd ) @ Ve )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
thf(fact_5450_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
! [V2: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_5451_vebt__delete_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,Tr: list @ vEBT_VEBT,Sm: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr @ Sm ) ) ).
% vebt_delete.simps(6)
thf(fact_5452_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
! [V2: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Vj )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
thf(fact_5453_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
! [V2: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Vi )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
thf(fact_5454_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
! [V2: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Vj )
= ( one_one @ nat ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
thf(fact_5455_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
! [V2: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Vi )
= ( one_one @ nat ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
thf(fact_5456_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_5457_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
thf(fact_5458_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
! [V2: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_5459_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
! [V2: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_5460_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_5461_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
thf(fact_5462_height__node,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% height_node
thf(fact_5463_size__list__conv__sum__list,axiom,
! [B: $tType] :
( ( size_list @ B )
= ( ^ [F3: B > nat,Xs2: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F3 @ Xs2 ) ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_5464_VEBT__internal_Oinsert__simp__mima,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
| ( X = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% VEBT_internal.insert_simp_mima
thf(fact_5465_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
! [V2: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_5466_VEBT__internal_Odelt__out__of__range,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: 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 @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% VEBT_internal.delt_out_of_range
thf(fact_5467_VEBT__internal_Otdeletemimi,axiom,
! [Deg: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ord_less_eq @ nat @ ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% VEBT_internal.tdeletemimi
thf(fact_5468_VEBT__internal_Otdeletemimi_H,axiom,
! [Deg: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ord_less_eq @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X ) @ ( one_one @ nat ) ) ) ).
% VEBT_internal.tdeletemimi'
thf(fact_5469_VEBT_Osize__gen_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X122: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X122 @ 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_5470_option_Osize_I4_J,axiom,
! [A: $tType,X22: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X22 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_5471_VEBT__internal_Odel__single__cont,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
& ( X = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% VEBT_internal.del_single_cont
thf(fact_5472_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_5473_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_5474_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
! [Uu3: nat,Uv3: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu3 @ Uv3 @ Uw2 ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_5475_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X222: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size_gen(2)
thf(fact_5476_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
! [V2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary ) @ X )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_5477_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,Va3: 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 ) @ Va3 @ Vb2 ) @ X3 ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: 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 @ V4 ) @ TreeList2 @ Vc2 ) @ X3 ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) @ X3 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_5478_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B6: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( zero_zero @ nat ) ) )
=> ( ! [A6: $o,B6: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A6: $o,B6: $o,N2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( suc @ ( suc @ N2 ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Uu2 ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: 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 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: 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 @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: 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 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
thf(fact_5479_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B6: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ 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 ) )
=> ( ! [V4: 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 ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ X3 ) )
=> ( ! [V4: 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 ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: 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 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_5480_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B6: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ X3 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) @ X3 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) @ X3 ) )
=> ( ! [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: 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 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_5481_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,B6: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B6 ) @ ( zero_zero @ nat ) ) )
=> ( ! [Uv2: $o,Uw: $o,N2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw ) @ ( suc @ N2 ) ) )
=> ( ! [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) @ Va3 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Ve2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Vi2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: 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 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_5482_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_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 ) ) )
=> ( ! [A6: $o,Uw: $o] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [A6: $o,B6: $o,Va: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( suc @ ( suc @ Va ) ) ) )
=> ( ! [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) @ Vb2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Vf2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Vj2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: 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 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_5483_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_5484_set__vebt__maxt_H,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) )
= ( ( member @ nat @ X @ ( vEBT_set_vebt @ T2 ) )
& ! [X2: nat] :
( ( member @ nat @ X2 @ ( vEBT_set_vebt @ T2 ) )
=> ( ord_less_eq @ nat @ X2 @ X ) ) ) ) ) ).
% set_vebt_maxt'
thf(fact_5485_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_i_n_t @ X )
= Y )
=> ( ! [A6: $o] :
( ? [B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A6 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ( Y
!= ( one_one @ 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
!= ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_5486_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X: A,N: nat,Y: A] :
( ( ( replicate @ A @ M @ X )
= ( replicate @ A @ N @ Y ) )
= ( ( M = N )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_5487_length__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_5488_map__replicate,axiom,
! [A: $tType,B: $tType,F2: B > A,N: nat,X: B] :
( ( map @ B @ A @ F2 @ ( replicate @ B @ N @ X ) )
= ( replicate @ A @ N @ ( F2 @ X ) ) ) ).
% map_replicate
thf(fact_5489_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_5490_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A2 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_5491_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_5492_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_5493_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_5494_set__vebt__maxt,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) )
= ( vEBT_VEBT_max_in_set @ ( vEBT_set_vebt @ T2 ) @ X ) ) ) ).
% set_vebt_maxt
thf(fact_5495_vebt__maxt_Osimps_I1_J,axiom,
! [B2: $o,A2: $o] :
( ( B2
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( A2
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_maxt.simps(1)
thf(fact_5496_VEBT__internal_Ointind,axiom,
! [A: $tType,I: nat,N: nat,P: A > $o,X: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( P @ X )
=> ( P @ ( nth @ A @ ( replicate @ A @ N @ X ) @ I ) ) ) ) ).
% VEBT_internal.intind
thf(fact_5497_replicate__length__same,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( X3 = X ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_5498_replicate__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate @ A @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5499_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_5500_map__replicate__const,axiom,
! [B: $tType,A: $tType,K: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X2: B] : K
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_5501_sum__list__replicate,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat,C2: A] :
( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N @ C2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ C2 ) ) ) ).
% sum_list_replicate
thf(fact_5502_vebt__inst_Oset__vebt__maxt_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ X ) )
= ( ( member @ nat @ X @ ( vEBT_set_vebt @ T2 ) )
& ! [X2: nat] :
( ( member @ nat @ X2 @ ( vEBT_set_vebt @ T2 ) )
=> ( ord_less_eq @ nat @ X2 @ X ) ) ) ) ) ).
% vebt_inst.set_vebt_maxt'
thf(fact_5503_vebt__maxt_Oelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ~ ( ( B6
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B6
=> ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( 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_5504_map__replicate__trivial,axiom,
! [A: $tType,X: A,I: nat] :
( ( map @ nat @ A
@ ^ [I4: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X ) ) ).
% map_replicate_trivial
thf(fact_5505_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_5506_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_5507_mint__bound,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ nat @ ( vEBT_T_m_i_n_t @ T2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% mint_bound
thf(fact_5508_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( vEBT_T_m_i_n_t @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A2 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
thf(fact_5509_vebt__buildup_Osimps_I3_J,axiom,
! [Va2: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_5510_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_5511_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_T_m_i_n_t @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ A6 @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A6 @ B6 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_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
= ( one_one @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_5512_vebt__maxt_Opelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( ( B6
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B6
=> ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A6 @ B6 ) ) ) )
=> ( ! [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_5513_VEBT__internal_Omax__in__set__def,axiom,
( vEBT_VEBT_max_in_set
= ( ^ [Xs2: set @ nat,X2: nat] :
( ( member @ nat @ X2 @ Xs2 )
& ! [Y4: nat] :
( ( member @ nat @ Y4 @ Xs2 )
=> ( ord_less_eq @ nat @ Y4 @ X2 ) ) ) ) ) ).
% VEBT_internal.max_in_set_def
thf(fact_5514_vebt__inst_Oset__vebt__maxt,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_set_vebt @ T2 ) @ X ) ) ) ).
% vebt_inst.set_vebt_maxt
thf(fact_5515_set__vebt__mint_H,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X ) )
= ( ( member @ nat @ X @ ( vEBT_set_vebt @ T2 ) )
& ! [X2: nat] :
( ( member @ nat @ X2 @ ( vEBT_set_vebt @ T2 ) )
=> ( ord_less_eq @ nat @ X @ X2 ) ) ) ) ) ).
% set_vebt_mint'
thf(fact_5516_VEBT__internal_Omintlistlength,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( Mi != Ma )
=> ( ( ord_less @ nat @ Mi @ Ma )
& ? [M4: nat] :
( ( ( some @ nat @ M4 )
= ( vEBT_vebt_mint @ Summary ) )
& ( ord_less @ nat @ M4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.mintlistlength
thf(fact_5517_option_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X22: A] :
( ( size_option @ A @ X @ ( some @ A @ X22 ) )
= ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_5518_vebt__minNull__mint,axiom,
( vEBT_VEBT_minNull
= ( ^ [T3: vEBT_VEBT] :
( ( vEBT_vebt_mint @ T3 )
= ( none @ nat ) ) ) ) ).
% vebt_minNull_mint
thf(fact_5519_vebt__inst_Oset__vebt__mint_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X ) )
= ( ( member @ nat @ X @ ( vEBT_set_vebt @ T2 ) )
& ! [X2: nat] :
( ( member @ nat @ X2 @ ( vEBT_set_vebt @ T2 ) )
=> ( ord_less_eq @ nat @ X @ X2 ) ) ) ) ) ).
% vebt_inst.set_vebt_mint'
thf(fact_5520_vebt__mint_Osimps_I1_J,axiom,
! [A2: $o,B2: $o] :
( ( A2
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( B2
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_mint.simps(1)
thf(fact_5521_VEBT__internal_Omisiz,axiom,
! [T2: vEBT_VEBT,N: nat,M: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( some @ nat @ M )
= ( vEBT_vebt_mint @ T2 ) )
=> ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% VEBT_internal.misiz
thf(fact_5522_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_5523_vebt__mint_Oelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X )
= Y )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ~ ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( ( B6
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B6
=> ( 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_5524_vebt__mint_Opelims,axiom,
! [X: vEBT_VEBT,Y: option @ nat] :
( ( ( vEBT_vebt_mint @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ X )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( ( B6
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B6
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A6 @ B6 ) ) ) )
=> ( ! [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_5525_set__vebt__mint,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ X ) )
= ( vEBT_VEBT_min_in_set @ ( vEBT_set_vebt @ T2 ) @ X ) ) ) ).
% set_vebt_mint
thf(fact_5526_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_5527_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_5528_VEBT__internal_Omin__in__set__def,axiom,
( vEBT_VEBT_min_in_set
= ( ^ [Xs2: set @ nat,X2: nat] :
( ( member @ nat @ X2 @ Xs2 )
& ! [Y4: nat] :
( ( member @ nat @ Y4 @ Xs2 )
=> ( ord_less_eq @ nat @ X2 @ Y4 ) ) ) ) ) ).
% VEBT_internal.min_in_set_def
thf(fact_5529_insersimp_H,axiom,
! [T2: vEBT_VEBT,N: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_12 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ Y ) @ ( one_one @ nat ) ) ) ) ).
% insersimp'
thf(fact_5530_insersimp,axiom,
! [T2: vEBT_VEBT,N: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_12 )
=> ( ord_less_eq @ nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ Y ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ).
% insersimp
thf(fact_5531_VEBT__internal_Ovalid__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 ) ) ) ).
% VEBT_internal.valid_insert_both_member_options_add
thf(fact_5532_VEBT__internal_Ovalid__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 ) ) ) ) ) ).
% VEBT_internal.valid_insert_both_member_options_pres
thf(fact_5533_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_5534_vebt__inst_Oset__vebt__mint,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_set_vebt @ T2 ) @ X ) ) ) ).
% vebt_inst.set_vebt_mint
thf(fact_5535_set__vebt__mint_H_H,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( ( ( vEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) ) )
& ( ( ( vEBT_set_vebt @ T2 )
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ ( lattic643756798350308766er_Min @ nat @ ( vEBT_set_vebt @ T2 ) ) ) ) ) ) ) ).
% set_vebt_mint''
thf(fact_5536_set__vebt__maxt_H_H,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( ( ( vEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( none @ nat ) ) )
& ( ( ( vEBT_set_vebt @ T2 )
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ ( lattic643756798349783984er_Max @ nat @ ( vEBT_set_vebt @ T2 ) ) ) ) ) ) ) ).
% set_vebt_maxt''
thf(fact_5537_VEBT__internal_Opower__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 ) ) ) ).
% VEBT_internal.power_shift
thf(fact_5538_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Max_singleton
thf(fact_5539_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Min_singleton
thf(fact_5540_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_5541_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_5542_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_5543_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_5544_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_5545_Min__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ X ) ) ) ) ).
% Min_le
thf(fact_5546_Min__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( member @ A @ X @ A3 )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= X ) ) ) ) ) ).
% Min_eqI
thf(fact_5547_Min_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A2 ) ) ) ) ).
% Min.coboundedI
thf(fact_5548_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A3 ) ) ) ) ).
% Min_in
thf(fact_5549_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ).
% Max_ge
thf(fact_5550_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( member @ A @ X @ A3 )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= X ) ) ) ) ) ).
% Max_eqI
thf(fact_5551_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ B3 )
& ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B3 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
& ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( lattic643756798349783984er_Max @ A @ B3 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_5552_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ A @ A2 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_5553_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ A3 ) ) ) ) ).
% Max_in
thf(fact_5554_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ A @ X @ A6 ) )
=> ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_5555_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A3 ) )
=> ! [A11: A] :
( ( member @ A @ A11 @ A3 )
=> ( ord_less_eq @ A @ X @ A11 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_5556_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( ( member @ A @ M @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ M @ X2 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_5557_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ X )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_5558_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A3 )
= M )
= ( ( member @ A @ M @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ M @ X2 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_5559_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ X )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_5560_Min__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [B6: A] :
( ( member @ A @ B6 @ A3 )
=> ( ord_less_eq @ A @ A2 @ B6 ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ A2 @ A3 ) )
= A2 ) ) ) ) ).
% Min_insert2
thf(fact_5561_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ A @ A6 @ X ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X ) ) ) ) ) ).
% Max.boundedI
thf(fact_5562_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X )
=> ! [A11: A] :
( ( member @ A @ A11 @ A3 )
=> ( ord_less_eq @ A @ A11 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_5563_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ( member @ A @ M @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_5564_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_5565_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A3 )
= M )
= ( ( member @ A @ M @ A3 )
& ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_5566_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_5567_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [B6: A] :
( ( member @ A @ B6 @ A3 )
=> ( ord_less_eq @ A @ B6 @ A2 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A2 @ A3 ) )
= A2 ) ) ) ) ).
% Max_insert2
thf(fact_5568_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B3 ) @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_5569_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M7: set @ A,N4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M7 @ N4 )
=> ( ( M7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N4 ) @ ( lattic643756798350308766er_Min @ A @ M7 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_5570_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B3 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_5571_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M7: set @ A,N4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M7 @ N4 )
=> ( ( M7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M7 ) @ ( lattic643756798349783984er_Max @ A @ N4 ) ) ) ) ) ) ).
% Max_mono
thf(fact_5572_card__le__Suc__Max,axiom,
! [S2: set @ nat] :
( ( finite_finite2 @ nat @ S2 )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S2 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S2 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_5573_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M3: nat,N3: nat] :
( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N3 ) @ M3 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_5574_vebt__inst_Oset__vebt__maxt_H_H,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( ( vEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( none @ nat ) ) )
& ( ( ( vEBT_set_vebt @ T2 )
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ( vEBT_vebt_maxt @ T2 )
= ( some @ nat @ ( lattic643756798349783984er_Max @ nat @ ( vEBT_set_vebt @ T2 ) ) ) ) ) ) ) ).
% vebt_inst.set_vebt_maxt''
thf(fact_5575_vebt__inst_Oset__vebt__mint_H_H,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ( ( vEBT_set_vebt @ T2 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( vEBT_vebt_mint @ T2 )
= ( none @ nat ) ) )
& ( ( ( vEBT_set_vebt @ T2 )
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ( vEBT_vebt_mint @ T2 )
= ( some @ nat @ ( lattic643756798350308766er_Min @ nat @ ( vEBT_set_vebt @ T2 ) ) ) ) ) ) ) ).
% vebt_inst.set_vebt_mint''
thf(fact_5576_VEBT__internal_Osetceilmax,axiom,
! [S: vEBT_VEBT,M: nat,Listy: list @ vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ S @ M )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ Listy ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( M
= ( suc @ N ) )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ Listy ) )
=> ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ X3 ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) )
=> ( ( ( semiring_1_of_nat @ int @ ( vEBT_VEBT_height @ S ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) )
=> ( ( semiring_1_of_nat @ int @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ S @ ( set2 @ vEBT_VEBT @ Listy ) ) ) ) )
= ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.setceilmax
thf(fact_5577_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ? [X6: A] :
( ( member @ A @ X6 @ ( set2 @ A @ Xs ) )
& ( P @ X6 ) )
=> ( ( find @ A @ P @ Xs )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_5578_invar__vebt_Ocases,axiom,
! [A12: vEBT_VEBT,A23: nat] :
( ( vEBT_invar_vebt @ A12 @ A23 )
=> ( ( ? [A6: $o,B6: $o] :
( A12
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( A23
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X6 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4 = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M4 ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X6 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M4 ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X6 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4 = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M4 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X6: nat] :
( ( ( ( vEBT_VEBT_high @ X6 @ N2 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X6 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X6 )
& ( ord_less_eq @ nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X6 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M4 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X6: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X6 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X6: nat] :
( ( ( ( vEBT_VEBT_high @ X6 @ N2 )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X6 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X6 )
& ( ord_less_eq @ nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_5579_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,X: B,A3: set @ B] :
( ( B2
= ( F2 @ X ) )
=> ( ( member @ B @ X @ A3 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_5580_image__ident,axiom,
! [A: $tType,Y6: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : X2
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_5581_image__empty,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_5582_empty__is__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F2 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_5583_image__is__empty,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( ( image @ B @ A @ F2 @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_5584_image__insert,axiom,
! [A: $tType,B: $tType,F2: B > A,A2: B,B3: set @ B] :
( ( image @ B @ A @ F2 @ ( insert @ B @ A2 @ B3 ) )
= ( insert @ A @ ( F2 @ A2 ) @ ( image @ B @ A @ F2 @ B3 ) ) ) ).
% image_insert
thf(fact_5585_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,F2: A > B] :
( ( member @ A @ X @ A3 )
=> ( ( insert @ B @ ( F2 @ X ) @ ( image @ A @ B @ F2 @ A3 ) )
= ( image @ A @ B @ F2 @ A3 ) ) ) ).
% insert_image
thf(fact_5586_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S2 )
= S2 ) ) ).
% image_add_0
thf(fact_5587_list_Oset__map,axiom,
! [B: $tType,A: $tType,F2: A > B,V2: list @ A] :
( ( set2 @ B @ ( map @ A @ B @ F2 @ V2 ) )
= ( image @ A @ B @ F2 @ ( set2 @ A @ V2 ) ) ) ).
% list.set_map
thf(fact_5588_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A
@ ^ [N3: A] : ( plus_plus @ A @ N3 @ K )
@ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_5589_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D4: A,A2: A,B2: A] :
( ( image @ A @ A
@ ^ [T3: A] : ( minus_minus @ A @ T3 @ D4 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A2 @ D4 ) @ ( minus_minus @ A @ B2 @ D4 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_5590_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image @ A @ A
@ ^ [N3: A] : ( plus_plus @ A @ N3 @ K )
@ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_5591_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: B > $o,F2: B > A,G: B > A,S2: set @ B] :
( ( image @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ S2 )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ S2 @ ( collect @ B @ P ) ) )
@ ( image @ B @ A @ G
@ ( inf_inf @ ( set @ B ) @ S2
@ ( collect @ B
@ ^ [X2: B] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_5592_Max__divisors__self__int,axiom,
! [N: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ N ) ) )
= ( abs_abs @ int @ N ) ) ) ).
% Max_divisors_self_int
thf(fact_5593_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ B,C2: A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [Uu: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% Max_const
thf(fact_5594_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ B,C2: A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image @ B @ A
@ ^ [Uu: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% Min_const
thf(fact_5595_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D4: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D4 )
=> ( ( image @ A @ A @ ( times_times @ A @ D4 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D4 @ A2 ) @ ( times_times @ A @ D4 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_5596_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D4: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D4 )
=> ( ( image @ A @ A
@ ^ [C6: A] : ( divide_divide @ A @ C6 @ D4 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A2 @ D4 ) @ ( divide_divide @ A @ B2 @ D4 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_5597_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S2 ) )
= ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S2 ) ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_5598_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S2 ) )
= ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S2 ) ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_5599_image__set,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( image @ B @ A @ F2 @ ( set2 @ B @ Xs ) )
= ( set2 @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ).
% image_set
thf(fact_5600_finite__surj,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ A @ B @ F2 @ A3 ) )
=> ( finite_finite2 @ B @ B3 ) ) ) ).
% finite_surj
thf(fact_5601_finite__subset__image,axiom,
! [A: $tType,B: $tType,B3: set @ A,F2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) )
=> ? [C8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C8 @ A3 )
& ( finite_finite2 @ B @ C8 )
& ( B3
= ( image @ B @ A @ F2 @ C8 ) ) ) ) ) ).
% finite_subset_image
thf(fact_5602_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] :
( ( ? [B4: set @ A] :
( ( finite_finite2 @ A @ B4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F2 @ A3 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set @ B] :
( ( finite_finite2 @ B @ B4 )
& ( ord_less_eq @ ( set @ B ) @ B4 @ A3 )
& ( P @ ( image @ B @ A @ F2 @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_5603_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] :
( ( ! [B4: set @ A] :
( ( ( finite_finite2 @ A @ B4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F2 @ A3 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set @ B] :
( ( ( finite_finite2 @ B @ B4 )
& ( ord_less_eq @ ( set @ B ) @ B4 @ A3 ) )
=> ( P @ ( image @ B @ A @ F2 @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_5604_image__Un,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ B] :
( ( image @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ ( image @ B @ A @ F2 @ B3 ) ) ) ).
% image_Un
thf(fact_5605_all__subset__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] :
( ( ! [B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F2 @ A3 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B4 @ A3 )
=> ( P @ ( image @ B @ A @ F2 @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_5606_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B,B3: set @ B] :
( ! [X3: A] :
( ( P @ X3 )
=> ( member @ B @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( collect @ A @ P ) ) @ B3 ) ) ).
% image_Collect_subsetI
thf(fact_5607_subset__image__iff,axiom,
! [A: $tType,B: $tType,B3: set @ A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A3 )
& ( B3
= ( image @ B @ A @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_5608_image__subset__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( member @ A @ ( F2 @ X2 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_5609_subset__imageE,axiom,
! [A: $tType,B: $tType,B3: set @ A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) )
=> ~ ! [C8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C8 @ A3 )
=> ( B3
!= ( image @ B @ A @ F2 @ C8 ) ) ) ) ).
% subset_imageE
thf(fact_5610_image__subsetI,axiom,
! [A: $tType,B: $tType,A3: set @ A,F2: A > B,B3: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ B @ ( F2 @ X3 ) @ B3 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 ) ) ).
% image_subsetI
thf(fact_5611_image__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B3 ) ) ) ).
% image_mono
thf(fact_5612_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ ( pow2 @ B @ A3 ) ) @ ( pow2 @ A @ B3 ) ) ) ).
% image_Pow_mono
thf(fact_5613_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A3 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A3 )
& ( ( F2 @ A5 )
= ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_5614_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,A3: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) )
=> ~ ! [X3: B] :
( ( B2
= ( F2 @ X3 ) )
=> ~ ( member @ B @ X3 @ A3 ) ) ) ).
% imageE
thf(fact_5615_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,G: C > B,A3: set @ C] :
( ( image @ B @ A @ F2 @ ( image @ C @ B @ G @ A3 ) )
= ( image @ C @ A
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) )
@ A3 ) ) ).
% image_image
thf(fact_5616_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( image @ B @ A @ F2 @ A3 ) )
& ( P @ X2 ) ) )
= ( image @ B @ A @ F2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_5617_imageI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,F2: A > B] :
( ( member @ A @ X @ A3 )
=> ( member @ B @ ( F2 @ X ) @ ( image @ A @ B @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_5618_image__iff,axiom,
! [A: $tType,B: $tType,Z3: A,F2: B > A,A3: set @ B] :
( ( member @ A @ Z3 @ ( image @ B @ A @ F2 @ A3 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( Z3
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_5619_bex__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
( ? [X6: A] :
( ( member @ A @ X6 @ ( image @ B @ A @ F2 @ A3 ) )
& ( P @ X6 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A3 )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_5620_image__cong,axiom,
! [B: $tType,A: $tType,M7: set @ A,N4: set @ A,F2: A > B,G: A > B] :
( ( M7 = N4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N4 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image @ A @ B @ F2 @ M7 )
= ( image @ A @ B @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_5621_ball__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F2 @ A3 ) )
=> ( P @ X3 ) )
=> ! [X6: B] :
( ( member @ B @ X6 @ A3 )
=> ( P @ ( F2 @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_5622_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,B2: B,F2: A > B] :
( ( member @ A @ X @ A3 )
=> ( ( B2
= ( F2 @ X ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_5623_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] :
( ( ( image @ B @ A @ F2 @ A3 )
= B3 )
=> ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ ( pow2 @ B @ A3 ) )
= ( pow2 @ A @ B3 ) ) ) ).
% image_Pow_surj
thf(fact_5624_find__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( find @ A @ P @ Xs )
= ( find @ A @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_5625_image__diff__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ ( image @ B @ A @ F2 @ B3 ) ) @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_5626_image__Int__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ ( image @ B @ A @ F2 @ B3 ) ) ) ).
% image_Int_subset
thf(fact_5627_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A3: set @ A,F7: B > A,F2: A > B,A9: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( F7 @ ( F2 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A9 )
=> ( ( F2 @ ( F7 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ A9 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F7 @ A9 ) @ A3 )
=> ( bij_betw @ A @ B @ F2 @ A3 @ A9 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_5628_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F2: A > B,A3: set @ A,A9: set @ B,B3: set @ A,B13: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A3 @ A9 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( ( image @ A @ B @ F2 @ B3 )
= B13 )
=> ( bij_betw @ A @ B @ F2 @ B3 @ B13 ) ) ) ) ).
% bij_betw_subset
thf(fact_5629_image__constant__conv,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X2: B] : C2
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X2: B] : C2
@ A3 )
= ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_5630_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A3: set @ A,C2: B] :
( ( member @ A @ X @ A3 )
=> ( ( image @ A @ B
@ ^ [X2: A] : C2
@ A3 )
= ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_5631_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,H2: B > A,G: B > C] :
( ( finite_finite2 @ B @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S2 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y4: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S2 )
& ( ( G @ X2 )
= Y4 ) ) ) )
@ ( image @ B @ C @ G @ S2 ) ) ) ) ) ).
% sum.image_gen
thf(fact_5632_translation__subtract__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= ( inf_inf @ ( set @ A )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ S )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_Int
thf(fact_5633_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ S )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_diff
thf(fact_5634_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,H2: B > A,G: B > C] :
( ( finite_finite2 @ B @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H2 @ S2 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y4: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S2 )
& ( ( G @ X2 )
= Y4 ) ) ) )
@ ( image @ B @ C @ G @ S2 ) ) ) ) ) ).
% prod.image_gen
thf(fact_5635_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_5636_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B,X: A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A3 )
=> ( ( F2 @ Y3 )
= ( F2 @ X ) ) )
=> ( ( the_elem @ B @ ( image @ A @ B @ F2 @ A3 ) )
= ( F2 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_5637_card__image__le,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image @ A @ B @ F2 @ A3 ) ) @ ( finite_card @ A @ A3 ) ) ) ).
% card_image_le
thf(fact_5638_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T5: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S2 ) @ T5 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y4: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S2 )
& ( ( G @ X2 )
= Y4 ) ) ) )
@ T5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S2 ) ) ) ) ) ) ).
% sum.group
thf(fact_5639_find__None__iff2,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( none @ A )
= ( find @ A @ P @ Xs ) )
= ( ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) ) ) ) ).
% find_None_iff2
thf(fact_5640_find__None__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( find @ A @ P @ Xs )
= ( none @ A ) )
= ( ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) ) ) ) ).
% find_None_iff
thf(fact_5641_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T5: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S2 ) @ T5 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y4: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S2 )
& ( ( G @ X2 )
= Y4 ) ) ) )
@ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S2 ) ) ) ) ) ) ).
% prod.group
thf(fact_5642_surj__card__le,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ A @ B @ F2 @ A3 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B3 ) @ ( finite_card @ A @ A3 ) ) ) ) ).
% surj_card_le
thf(fact_5643_scaleR__image__atLeastAtMost,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,X: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( image @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ Y ) ) ) ) ) ).
% scaleR_image_atLeastAtMost
thf(fact_5644_VEBT__internal_Oboth__member__options__ding,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% VEBT_internal.both_member_options_ding
thf(fact_5645_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S2: set @ B,F2: B > A,K: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F2 @ X2 ) @ K )
@ S2 ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image @ B @ A @ F2 @ S2 ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_5646_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S2: set @ B,F2: B > A,K: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F2 @ X2 ) @ K )
@ S2 ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image @ B @ A @ F2 @ S2 ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_5647_VEBT__internal_Oheight__compose__list,axiom,
! [T2: vEBT_VEBT,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ T2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_height @ T2 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary @ ( set2 @ vEBT_VEBT @ TreeList ) ) ) ) ) ) ).
% VEBT_internal.height_compose_list
thf(fact_5648_VEBT__internal_Oboth__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
| ( X = Mi )
| ( X = Ma ) ) ) ) ).
% VEBT_internal.both_member_options_from_complete_tree_to_child
thf(fact_5649_VEBT__internal_Ohigh__def,axiom,
( vEBT_VEBT_high
= ( ^ [X2: nat,N3: nat] : ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% VEBT_internal.high_def
thf(fact_5650_VEBT__internal_Olow__def,axiom,
( vEBT_VEBT_low
= ( ^ [X2: nat,N3: nat] : ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% VEBT_internal.low_def
thf(fact_5651_VEBT__internal_Oboth__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% VEBT_internal.both_member_options_from_chilf_to_complete_tree
thf(fact_5652_VEBT__internal_Ohigh__bound__aux,axiom,
! [Ma: nat,N: nat,M: nat] :
( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% VEBT_internal.high_bound_aux
thf(fact_5653_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X ) @ ( times_times @ A @ C2 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_5654_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,C2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_5655_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_5656_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_5657_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_5658_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_5659_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S2: set @ A,R2: set @ B,G: A > B,F2: B > C] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ B @ R2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G @ S2 ) @ R2 )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X2: A] : ( F2 @ ( G @ X2 ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y4: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S2 )
& ( ( G @ X2 )
= Y4 ) ) ) ) )
@ ( F2 @ Y4 ) )
@ R2 ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_5660_sum__le__card__Max,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( times_times @ nat @ ( finite_card @ A @ A3 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ A @ nat @ F2 @ A3 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_5661_find__Some__iff2,axiom,
! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
( ( ( some @ A @ X )
= ( find @ A @ P @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I4 ) )
& ( X
= ( nth @ A @ Xs @ I4 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I4 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_5662_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I4 ) )
& ( X
= ( nth @ A @ Xs @ I4 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I4 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_5663_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_5664_VEBT__internal_Olow__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 ) ) ).
% VEBT_internal.low_inv
thf(fact_5665_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_5666_VEBT__internal_Ohigh__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 ) ) ).
% VEBT_internal.high_inv
thf(fact_5667_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_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 )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_5668_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( X = Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( X = Ma ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma @ X ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
thf(fact_5669_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( X = Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( X = Ma ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi @ X )
& ( ord_less @ nat @ X @ Ma ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
@ ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
thf(fact_5670_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
thf(fact_5671_card__Min__le__sum,axiom,
! [A: $tType,A3: set @ A,F2: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A3 ) @ ( lattic643756798350308766er_Min @ nat @ ( image @ A @ nat @ F2 @ A3 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) ) ) ).
% card_Min_le_sum
thf(fact_5672_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_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 )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_5673_VEBT__internal_Omax__ins__scaled,axiom,
! [N: nat,X14: vEBT_VEBT,M: nat,X13: list @ vEBT_VEBT] : ( ord_less_eq @ nat @ ( times_times @ nat @ N @ ( vEBT_VEBT_height @ X14 ) ) @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N @ ( lattic643756798349783984er_Max @ nat @ ( insert @ nat @ ( vEBT_VEBT_height @ X14 ) @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( set2 @ vEBT_VEBT @ X13 ) ) ) ) ) ) ) ).
% VEBT_internal.max_ins_scaled
thf(fact_5674_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa )
= Y )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( Xa = Ma2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi2 @ Xa )
& ( ord_less @ nat @ Xa @ Ma2 ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
@ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
thf(fact_5675_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa )
= Y )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( Xa
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( Xa = Ma2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
thf(fact_5676_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa )
= Y )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
thf(fact_5677_VEBT__internal_Oheight_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_height @ X )
= Y )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [Uu2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.height.elims
thf(fact_5678_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A13: vEBT_VEBT,A24: nat] :
( ( ? [A5: $o,B5: $o] :
( A13
= ( vEBT_Leaf @ A5 @ B5 ) )
& ( A24
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
( ( A13
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A24 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ N3 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
& ( A24
= ( plus_plus @ nat @ N3 @ N3 ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
( ( A13
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A24 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
& ( A24
= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) )
& ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A13
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A24 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ N3 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
& ( A24
= ( plus_plus @ nat @ N3 @ N3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A24 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A13
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A24 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
& ( A24
= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
=> ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A24 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_5679_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) @ Xa ) ) ) )
=> ( ! [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5076183648494686801_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_5680_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( Xa
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( Xa = Ma2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_e_m_b_e_r @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T5837161174952499735_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_5681_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( Xa = Ma2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( zero_zero @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( zero_zero @ nat )
@ ( if @ nat
@ ( ( ord_less @ nat @ Mi2 @ Xa )
& ( ord_less @ nat @ Xa @ Ma2 ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( zero_zero @ nat ) )
@ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8099345112685741742_r_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_5682_bij__betw__Suc,axiom,
! [M7: set @ nat,N4: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M7 @ N4 )
= ( ( image @ nat @ nat @ suc @ M7 )
= N4 ) ) ).
% bij_betw_Suc
thf(fact_5683_bij__betw__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N4: set @ nat,A3: set @ A] :
( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N4 @ A3 )
= ( ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ N4 )
= A3 ) ) ) ).
% bij_betw_of_nat
thf(fact_5684_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set @ ( product_prod @ A @ B ),F2: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A3 )
=> ( member @ C @ ( F2 @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F2 ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_5685_zero__notin__Suc__image,axiom,
! [A3: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A3 ) ) ).
% zero_notin_Suc_image
thf(fact_5686_nat__seg__image__imp__finite,axiom,
! [A: $tType,A3: set @ A,F2: nat > A,N: nat] :
( ( A3
= ( image @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) ) )
=> ( finite_finite2 @ A @ A3 ) ) ).
% nat_seg_image_imp_finite
thf(fact_5687_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A4: set @ A] :
? [N3: nat,F3: nat > A] :
( A4
= ( image @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N3 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_5688_finite__int__iff__bounded,axiom,
( ( finite_finite2 @ int )
= ( ^ [S6: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S6 ) @ ( set_ord_lessThan @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_5689_finite__int__iff__bounded__le,axiom,
( ( finite_finite2 @ int )
= ( ^ [S6: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S6 ) @ ( set_ord_atMost @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_5690_in__image__insert__iff,axiom,
! [A: $tType,B3: set @ ( set @ A ),X: A,A3: set @ A] :
( ! [C8: set @ A] :
( ( member @ ( set @ A ) @ C8 @ B3 )
=> ~ ( member @ A @ X @ C8 ) )
=> ( ( member @ ( set @ A ) @ A3 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X ) @ B3 ) )
= ( ( member @ A @ X @ A3 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_5691_image__int__atLeastAtMost,axiom,
! [A2: nat,B2: nat] :
( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastAtMost
thf(fact_5692_image__int__atLeastLessThan,axiom,
! [A2: nat,B2: nat] :
( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) )
= ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastLessThan
thf(fact_5693_Pow__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( pow2 @ A @ ( insert @ A @ A2 @ A3 ) )
= ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ A2 ) @ ( pow2 @ A @ A3 ) ) ) ) ).
% Pow_insert
thf(fact_5694_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_5695_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_5696_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_5697_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_5698_subset__subseqs,axiom,
! [A: $tType,X5: set @ A,Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( set2 @ A @ Xs ) )
=> ( member @ ( set @ A ) @ X5 @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_5699_subseqs__powset,axiom,
! [A: $tType,Xs: list @ A] :
( ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
= ( pow2 @ A @ ( set2 @ A @ Xs ) ) ) ).
% subseqs_powset
thf(fact_5700_image__add__int__atLeastLessThan,axiom,
! [L: int,U2: int] :
( ( image @ int @ int
@ ^ [X2: int] : ( plus_plus @ int @ X2 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U2 @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U2 ) ) ).
% image_add_int_atLeastLessThan
thf(fact_5701_image__atLeastZeroLessThan__int,axiom,
! [U2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U2 )
=> ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U2 )
= ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U2 ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_5702_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C2 @ Y )
=> ( ( image @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y )
=> ( ( ( ord_less @ nat @ X @ Y )
=> ( ( image @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_5703_VEBT__internal_Oheight_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( ( vEBT_VEBT_height @ X )
= Y )
=> ( ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ X )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Y
= ( zero_zero @ nat ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A6 @ B6 ) ) ) )
=> ~ ! [Uu2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( insert @ vEBT_VEBT @ Summary2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) ) ) ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% VEBT_internal.height.pelims
thf(fact_5704_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa )
= Y )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( Xa
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
thf(fact_5705_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
thf(fact_5706_minNull__bound,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ nat @ ( vEBT_T_m_i_n_N_u_l_l @ T2 ) @ ( one_one @ nat ) ) ).
% minNull_bound
thf(fact_5707_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( Xa
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) @ Xa ) ) ) )
=> ( ! [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T9217963907923527482_t_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_5708_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa )
= Y )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
thf(fact_5709_vebt__delete_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X @ Xa )
= Y )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ B6 ) ) ) )
=> ( ! [A6: $o] :
( ? [B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ A6 @ $false ) ) ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( Y
!= ( vEBT_Leaf @ A6 @ B6 ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ 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 @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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 @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.elims
thf(fact_5710_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_5711_Min_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Min.infinite
thf(fact_5712_VEBT__internal_Osummaxma,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi != Ma )
=> ( ( the2 @ nat @ ( vEBT_vebt_maxt @ Summary ) )
= ( vEBT_VEBT_high @ Ma @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% VEBT_internal.summaxma
thf(fact_5713_VEBT__internal_Onested__mint,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va2: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( N
= ( suc @ ( suc @ Va2 ) ) )
=> ( ~ ( 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 @ Va2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide @ nat @ Va2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ).
% VEBT_internal.nested_mint
thf(fact_5714_VEBT__internal_Odel__x__not__mi__newnode__not__nil,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: 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 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( 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 @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.del_x_not_mi_newnode_not_nil
thf(fact_5715_VEBT__internal_Odel__x__mi__lets__in__not__minNull,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: 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 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( 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 @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.del_x_mi_lets_in_not_minNull
thf(fact_5716_VEBT__internal_Odel__x__not__mi,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: 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 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ 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 @ H2 ) )
= ( 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 @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
& ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( 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 @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.del_x_not_mi
thf(fact_5717_VEBT__internal_Odel__x__not__mia,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: 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 ) ) ) )
= H2 )
=> ( ( ( 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 ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ 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 @ H2 ) )
= ( 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 @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
@ ( vEBT_vebt_delete @ Summary @ H2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( if @ nat @ ( X = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.del_x_not_mia
thf(fact_5718_VEBT__internal_Odel__x__not__mi__new__node__nil,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: 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 ) ) ) )
= H2 )
=> ( ( ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H2 ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.del_x_not_mi_new_node_nil
thf(fact_5719_VEBT__internal_Odel__x__mi__lets__in__minNull,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: 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 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( Sn
= ( vEBT_vebt_delete @ Summary @ H2 ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.del_x_mi_lets_in_minNull
thf(fact_5720_VEBT__internal_Odel__x__mi__lets__in,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: 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 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( Newnode
= ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
=> ( ( Newlist
= ( list_update @ vEBT_VEBT @ TreeList @ H2 @ Newnode ) )
=> ( ( ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ Xn
@ ( if @ nat @ ( Xn = Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
= ( 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 @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ Newlist @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ Newlist
@ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
& ( ~ ( vEBT_VEBT_minNull @ Newnode )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( 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 @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.del_x_mi_lets_in
thf(fact_5721_VEBT__internal_Odel__in__range,axiom,
! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: 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 @ TreeList @ 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ Summary ) ) ) ) ) ) ).
% VEBT_internal.del_in_range
thf(fact_5722_VEBT__internal_Odel__x__mia,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: 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 @ TreeList @ 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( if @ nat
@ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
@ ( if @ nat
@ ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).
% VEBT_internal.del_x_mia
thf(fact_5723_VEBT__internal_Odel__x__mi,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: 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 ) ) ) )
= H2 )
=> ( ( Xn
= ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
=> ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= L )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ 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 @ H2 ) )
= ( 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 @ H2 ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) )
@ ( vEBT_vebt_delete @ Summary @ H2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xn @ ( if @ nat @ ( Xn = Ma ) @ ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.del_x_mi
thf(fact_5724_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
=> ( ( ( ( X = Mi )
& ( X = Ma ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ( X = Mi )
& ( X = Ma ) )
=> ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
thf(fact_5725_vebt__delete_Osimps_I7_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: 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 @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ 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 @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
& ( ~ ( ( X = Mi )
& ( X = Ma ) )
=> ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va2 ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma ) ) )
@ ( suc @ ( suc @ Va2 ) )
@ ( list_update @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).
% vebt_delete.simps(7)
thf(fact_5726_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) @ ( plus_plus @ nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V6368547301243506412_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
thf(fact_5727_vebt__delete_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_delete @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ B6 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ A6 @ $false ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TrLst2: list @ vEBT_VEBT,Smry2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TrLst2 @ Smry2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Tr2: list @ vEBT_VEBT,Sm2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_delete_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ Tr2 @ Sm2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
=> ( ( ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
& ( ~ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
=> ( Y
= ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ 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 @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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 @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( Xa = Mi2 ) @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
@ ( if @ nat
@ ( ( ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma2 ) ) )
@ ( suc @ ( suc @ Va ) )
@ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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 ) ) @ TreeList2 @ 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_delete.pelims
thf(fact_5728_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa )
= Y )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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 @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
thf(fact_5729_maxt__bound,axiom,
! [T2: vEBT_VEBT] : ( ord_less_eq @ nat @ ( vEBT_T_m_a_x_t @ T2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ).
% maxt_bound
thf(fact_5730_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ X @ Mi )
| ( ord_less @ nat @ Ma @ X ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( X = Mi )
& ( X = Ma ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( X = Mi ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_d_e_l_e_t_e @ 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat )
@ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma ) )
& ( ( X != Mi )
=> ( X = Ma ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
thf(fact_5731_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ ( suc @ N2 ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
=> ( ! [Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( zero_zero @ nat ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ord_less @ nat @ Xa @ Mi2 )
| ( ord_less @ nat @ Ma2 @ Xa ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( Xa = Mi2 )
& ( Xa = Ma2 ) )
@ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) @ ( if @ nat @ ( Xa = Mi2 ) @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) @ ( one_one @ nat ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( if @ nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( 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 @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( if @ nat
@ ( ( ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
= Ma2 ) )
& ( ( Xa != Mi2 )
=> ( Xa = Ma2 ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( Xa = 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 @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T8441311223069195367_e_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_5732_vebt__delete__code_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_delete @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( case_option @ vEBT_VEBT @ ( product_prod @ nat @ nat ) @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary )
@ ^ [Mima: product_prod @ nat @ nat] :
( if @ vEBT_VEBT @ ( ord_less_eq @ nat @ Deg @ ( one_one @ nat ) ) @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary )
@ ( product_case_prod @ nat @ nat @ vEBT_VEBT
@ ^ [Mi3: nat,Ma3: nat] :
( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ X @ Mi3 )
| ( ord_less @ nat @ Ma3 @ X ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg @ TreeList @ Summary )
@ ( if @ vEBT_VEBT
@ ( ( X = Mi3 )
& ( X = Ma3 ) )
@ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary )
@ ( if @ vEBT_VEBT @ ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi3 ) @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi3 )
@ ( if @ nat
@ ( ( ( X = Mi3 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma3 ) )
& ( ( X != Mi3 )
=> ( X = Ma3 ) ) )
@ ( if @ nat
@ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
= ( none @ nat ) )
@ ( if @ nat @ ( X = Mi3 ) @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi3 )
@ ( 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 = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ 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 = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) )
@ Ma3 ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( vEBT_Node
@ ( some @ ( product_prod @ nat @ nat )
@ ( product_Pair @ nat @ nat @ ( if @ nat @ ( X = Mi3 ) @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi3 )
@ ( if @ nat
@ ( ( ( X = Mi3 )
=> ( ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
= Ma3 ) )
& ( ( X != Mi3 )
=> ( X = Ma3 ) ) )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ 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 @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ Ma3 ) ) )
@ Deg
@ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_delete @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( X = Mi3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( the2 @ nat @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg @ TreeList @ Summary ) ) ) )
@ Mima ) )
@ Info ) ) ).
% vebt_delete_code(2)
thf(fact_5733_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ! [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] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_5734_option_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H2: B > C,F1: B,F22: A > B,Option: option @ A] :
( ( H2 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( case_option @ C @ A @ ( H2 @ F1 )
@ ^ [X2: A] : ( H2 @ ( F22 @ X2 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_5735_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_5736_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va2: 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 ) @ Va2 @ Vb ) @ X )
= ( ( X = Mi )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_5737_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V2: nat,TreeList: list @ vEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_5738_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V2: nat,TreeList: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList @ Vc ) @ X )
= ( ( X = Mi )
| ( X = Ma )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_5739_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_5740_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa )
= 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] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( Y
= ( ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) )
=> ( Y
= ( ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_5741_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [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 ) @ Xa ) ) ) )
=> ( ! [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 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Y
= ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) )
=> ( ( Y
= ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_5742_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [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 ) @ Xa ) ) )
=> ( ! [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 ) @ Xa ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) @ Xa ) )
=> ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_5743_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) @ Xa ) )
=> ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd2 ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_5744_option_Odisc__eq__case_I1_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
= ( none @ A ) )
= ( case_option @ $o @ A @ $true
@ ^ [Uu: A] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_5745_option_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
= ( case_option @ $o @ A @ $false
@ ^ [Uu: A] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_5746_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X )
=> ( ( ( X
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y3: A] :
( ( X
= ( some @ A @ Y3 ) )
=> ~ ( Q @ Y3 ) ) ) ) ).
% case_optionE
thf(fact_5747_vebt__insert__code_I2_J,axiom,
! [Deg: nat,Info: option @ ( product_prod @ nat @ nat ),TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ( ord_less_eq @ nat @ Deg @ ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) )
& ( ~ ( ord_less_eq @ nat @ Deg @ ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( case_option @ vEBT_VEBT @ ( product_prod @ nat @ nat ) @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ X ) ) @ Deg @ TreeList @ Summary )
@ ( product_case_prod @ nat @ nat @ vEBT_VEBT
@ ^ [Mi3: nat,Ma3: nat] :
( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi3 )
| ( X = Ma3 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ X @ Mi3 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ Ma3 ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi3 ) @ Mi3 @ X ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg @ TreeList @ Summary ) ) )
@ Info ) ) ) ) ).
% vebt_insert_code(2)
thf(fact_5748_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa )
= Y )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( Y
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B6 )
& ( Xa
= ( 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 ),V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S3 ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_5749_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B6 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S3 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_5750_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% max.bounded_iff
thf(fact_5751_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb2
thf(fact_5752_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb1
thf(fact_5753_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb3
thf(fact_5754_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb4
thf(fact_5755_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y ) @ Z3 )
= ( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Y @ Z3 ) ) ) ) ).
% max_less_iff_conj
thf(fact_5756_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% max_bot
thf(fact_5757_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% max_bot2
thf(fact_5758_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ M @ N ) ) ) ).
% max_Suc_Suc
thf(fact_5759_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_5760_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_5761_max__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% max_nat.right_neutral
thf(fact_5762_max__nat_Oneutr__eq__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( ord_max @ nat @ A2 @ B2 ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_5763_max__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% max_nat.left_neutral
thf(fact_5764_max__nat_Oeq__neutr__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_max @ nat @ A2 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_5765_of__bool__or__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
| Q ) )
= ( ord_max @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_or_iff
thf(fact_5766_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_5767_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_5768_max__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U2: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U2 ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U2 ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ U2 ) ) ) ) ) ).
% max_number_of(1)
thf(fact_5769_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_5770_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_5771_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_5772_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_5773_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U2: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( numeral_numeral @ A @ U2 ) ) ) ) ) ).
% max_number_of(2)
thf(fact_5774_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U2: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) ) ) ) ) ).
% max_number_of(3)
thf(fact_5775_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U2: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U2 ) ) ) ) ) ) ).
% max_number_of(4)
thf(fact_5776_max__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_max @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_max @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_5777_max__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_5778_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A3 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ).
% Max_insert
thf(fact_5779_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ B5 @ A5 ) ) ) ) ).
% max_def
thf(fact_5780_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_5781_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_5782_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B5 ) @ B5 @ A5 ) ) ) ) ).
% max_def_raw
thf(fact_5783_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_5784_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_5785_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_max @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% max.absorb_iff2
thf(fact_5786_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_max @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% max.absorb_iff1
thf(fact_5787_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_5788_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.cobounded2
thf(fact_5789_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.cobounded1
thf(fact_5790_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( A5
= ( ord_max @ A @ A5 @ B5 ) ) ) ) ) ).
% max.order_iff
thf(fact_5791_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% max.boundedI
thf(fact_5792_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% max.boundedE
thf(fact_5793_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( ord_max @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% max.orderI
thf(fact_5794_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.orderE
thf(fact_5795_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,D4: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D4 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D4 ) @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_5796_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_5797_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_5798_nat__add__max__left,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N ) @ Q3 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q3 ) @ ( plus_plus @ nat @ N @ Q3 ) ) ) ).
% nat_add_max_left
thf(fact_5799_nat__add__max__right,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N @ Q3 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ N ) @ ( plus_plus @ nat @ M @ Q3 ) ) ) ).
% nat_add_max_right
thf(fact_5800_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_5801_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_5802_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less @ A @ Z3 @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z3 @ X )
| ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% less_max_iff_disj
thf(fact_5803_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% max.strict_boundedE
thf(fact_5804_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( A5
= ( ord_max @ A @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_5805_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ A2 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_5806_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_5807_of__int__max,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,Y: int] :
( ( ring_1_of_int @ A @ ( ord_max @ int @ X @ Y ) )
= ( ord_max @ A @ ( ring_1_of_int @ A @ X ) @ ( ring_1_of_int @ A @ Y ) ) ) ) ).
% of_int_max
thf(fact_5808_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_5809_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ M @ ( ord_max @ nat @ N @ Q3 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q3 ) ) ) ).
% nat_mult_max_right
thf(fact_5810_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M @ N ) @ Q3 )
= ( ord_max @ nat @ ( times_times @ nat @ M @ Q3 ) @ ( times_times @ nat @ N @ Q3 ) ) ) ).
% nat_mult_max_left
thf(fact_5811_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ N @ M ) @ M )
= ( ord_max @ nat @ N @ M ) ) ).
% nat_minus_add_max
thf(fact_5812_Max_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ).
% Max.in_idem
thf(fact_5813_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu3: option @ ( product_prod @ nat @ nat ),Uv3: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu3 @ ( zero_zero @ nat ) @ Uv3 @ Uw2 ) @ Ux2 ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_5814_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N4: set @ A] :
( ! [X3: A,Y3: A] :
( ( H2 @ ( ord_max @ A @ X3 @ Y3 ) )
= ( ord_max @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798349783984er_Max @ A @ N4 ) )
= ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ H2 @ N4 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_5815_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B3 ) @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ).
% Max.subset
thf(fact_5816_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A3 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_5817_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( ord_max @ A @ X3 @ Y3 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Max.closed
thf(fact_5818_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B3 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_5819_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B2 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_5820_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_5821_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A3 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_5822_VEBT__internal_Oheight__i__max,axiom,
! [I: nat,X13: list @ vEBT_VEBT,Foo: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ X13 ) )
=> ( ord_less_eq @ nat @ ( vEBT_VEBT_height @ ( nth @ vEBT_VEBT @ X13 @ I ) ) @ ( ord_max @ nat @ Foo @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( set2 @ vEBT_VEBT @ X13 ) ) ) ) ) ) ).
% VEBT_internal.height_i_max
thf(fact_5823_VEBT__internal_Omax__idx__list,axiom,
! [I: nat,X13: list @ vEBT_VEBT,N: nat,X14: vEBT_VEBT] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ vEBT_VEBT ) @ X13 ) )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ N @ ( vEBT_VEBT_height @ ( nth @ vEBT_VEBT @ X13 @ I ) ) ) @ ( suc @ ( suc @ ( times_times @ nat @ N @ ( ord_max @ nat @ ( vEBT_VEBT_height @ X14 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ vEBT_VEBT @ nat @ vEBT_VEBT_height @ ( set2 @ vEBT_VEBT @ X13 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.max_idx_list
thf(fact_5824_VEBT__internal_Oinsert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList: list @ vEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ X @ Mi )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ ( ord_max @ nat @ Mi @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% VEBT_internal.insert_simp_excp
thf(fact_5825_VEBT__internal_Oinsert__simp__norm,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ Mi @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( ord_max @ nat @ X @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% VEBT_internal.insert_simp_norm
thf(fact_5826_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_5827_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy2: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_5828_vebt__insert_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa )
= Y )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B6 ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A6 @ $true ) ) )
& ( ( Xa
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A6 @ B6 ) ) ) ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) ) )
=> ( ! [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_5829_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B6 )
& ( Xa
= ( 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 ),V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S3 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_5830_vebt__insert_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B6 ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A6 @ $true ) ) )
& ( ( Xa
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A6 @ B6 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) @ Xa ) ) ) )
=> ( ! [V4: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V4 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa = Mi2 )
| ( Xa = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( 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 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_5831_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Y
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B6 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) ) ) )
=> ( ! [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 ) @ Xa ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S3 ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_5832_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B6 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S3 ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_5833_max__enat__simps_I2_J,axiom,
! [Q3: extended_enat] :
( ( ord_max @ extended_enat @ Q3 @ ( zero_zero @ extended_enat ) )
= Q3 ) ).
% max_enat_simps(2)
thf(fact_5834_max__enat__simps_I3_J,axiom,
! [Q3: extended_enat] :
( ( ord_max @ extended_enat @ ( zero_zero @ extended_enat ) @ Q3 )
= Q3 ) ).
% max_enat_simps(3)
thf(fact_5835_sup__int__def,axiom,
( ( sup_sup @ int )
= ( ord_max @ int ) ) ).
% sup_int_def
thf(fact_5836_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B6 )
& ( Xa
= ( 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 ) @ Xa ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S3 ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_5837_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B6 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_5838_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa )
= Y )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( Y
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B6 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw ) )
=> Y )
=> ( ( ? [V4: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> Y )
=> ( ( ? [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> Y )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_5839_set__vebt__member,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( vEBT_vebt_member @ T2 @ X )
= ( member @ nat @ X @ ( vEBT_set_vebt @ T2 ) ) ) ) ).
% set_vebt_member
thf(fact_5840_sup__enat__def,axiom,
( ( sup_sup @ extended_enat )
= ( ord_max @ extended_enat ) ) ).
% sup_enat_def
thf(fact_5841_VEBT__internal_Oset__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% VEBT_internal.set_vebt'_def
thf(fact_5842_vebt__member__code_I2_J,axiom,
! [T2: nat,R3: list @ vEBT_VEBT,E2: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ T2 @ R3 @ E2 ) @ X ) ).
% vebt_member_code(2)
thf(fact_5843_vebt__inst_Oset__vebt__member,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 ) ) ) ) ).
% vebt_inst.set_vebt_member
thf(fact_5844_VEBT__internal_Ogreater__shift,axiom,
( ( ord_less @ nat )
= ( ^ [Y4: nat,X2: nat] : ( vEBT_VEBT_greater @ ( some @ nat @ X2 ) @ ( some @ nat @ Y4 ) ) ) ) ).
% VEBT_internal.greater_shift
thf(fact_5845_vebt__member__code_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B2 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% vebt_member_code(1)
thf(fact_5846_vebt__member_Osimps_I3_J,axiom,
! [V2: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz ) @ X ) ).
% vebt_member.simps(3)
thf(fact_5847_vebt__member_Osimps_I4_J,axiom,
! [V2: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X ) ).
% vebt_member.simps(4)
thf(fact_5848_VEBT__internal_Omint__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 ) ) ) ) ).
% VEBT_internal.mint_corr_help
thf(fact_5849_VEBT__internal_Omaxt__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 ) ) ) ) ).
% VEBT_internal.maxt_corr_help
thf(fact_5850_VEBT__internal_Omember__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 ) ) ) ) ).
% VEBT_internal.member_bound
thf(fact_5851_VEBT__internal_Opost__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 ) ) ) ) ) ) ).
% VEBT_internal.post_member_pre_member
thf(fact_5852_VEBT__internal_Omember__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X = Mi )
| ( X = Ma )
| ( ( ord_less @ nat @ X @ Ma )
& ( ord_less @ nat @ Mi @ X )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.member_inv
thf(fact_5853_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( ( X != Mi )
=> ( ( X != Ma )
=> ( ~ ( ord_less @ nat @ X @ Mi )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ~ ( ord_less @ nat @ Ma @ X )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_5854_vebt__member__code_I3_J,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( ~ ( ( Deg
= ( zero_zero @ nat ) )
| ( Deg
= ( suc @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( Deg
= ( zero_zero @ nat ) )
| ( Deg
= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ( 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 @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member_code(3)
thf(fact_5855_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X @ Xa )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B6 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_5856_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ A6 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ? [Va: nat] :
( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B6 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) ) ) )
=> ( ( ? [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
thf(fact_5857_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( Y
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B6 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) ) ) )
=> ( ! [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 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( 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 ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( 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 ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_5858_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B6 )
& ( Xa
= ( 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 ) @ Xa ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( 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 ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ Xa ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( 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 ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
=> ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_5859_set__vebt__pred,axiom,
! [T2: vEBT_VEBT,X: nat,Px: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat @ Px ) )
= ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T2 ) @ X @ Px ) ) ) ).
% set_vebt_pred
thf(fact_5860_vebt__pred_Osimps_I1_J,axiom,
! [Uu3: $o,Uv3: $o] :
( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu3 @ Uv3 ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ).
% vebt_pred.simps(1)
thf(fact_5861_vebt__pred_Osimps_I5_J,axiom,
! [V2: product_prod @ nat @ nat,Vd: list @ vEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vd @ Ve ) @ Vf )
= ( none @ nat ) ) ).
% vebt_pred.simps(5)
thf(fact_5862_vebt__pred_Osimps_I6_J,axiom,
! [V2: product_prod @ nat @ nat,Vh: list @ vEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh @ Vi ) @ Vj )
= ( none @ nat ) ) ).
% vebt_pred.simps(6)
thf(fact_5863_vebt__pred_Osimps_I2_J,axiom,
! [A2: $o,Uw2: $o] :
( ( A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( none @ nat ) ) ) ) ).
% vebt_pred.simps(2)
thf(fact_5864_vebt__pred__code,axiom,
! [X: nat,A2: $o,B2: $o] :
( ( ( X
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( none @ nat ) ) )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> ( ( A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( none @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ nat ) )
=> ( ( B2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( none @ nat ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred_code
thf(fact_5865_VEBT__internal_Opred__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 ) ) ) ) ) ).
% VEBT_internal.pred_empty
thf(fact_5866_vebt__pred_Osimps_I3_J,axiom,
! [B2: $o,A2: $o,Va2: nat] :
( ( B2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) )
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A2
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) )
= ( none @ nat ) ) ) ) ) ) ).
% vebt_pred.simps(3)
thf(fact_5867_VEBT__internal_Ohelpypredd,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 ) ) ) ) ).
% VEBT_internal.helpypredd
thf(fact_5868_VEBT__internal_Opred__max,axiom,
! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: 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 @ TreeList @ Summary ) @ X )
= ( some @ nat @ Ma ) ) ) ) ).
% VEBT_internal.pred_max
thf(fact_5869_VEBT__internal_Opred__list__to__short,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X @ Ma )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( 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 @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) ) ) ) ).
% VEBT_internal.pred_list_to_short
thf(fact_5870_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
thf(fact_5871_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ A6 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [A6: $o,B6: $o] :
( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ? [Va: nat] :
( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
thf(fact_5872_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma @ X ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
thf(fact_5873_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B6 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
=> ~ ( ( Xa != Mi2 )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_5874_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ! [Va: nat] :
( ( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( if @ nat @ B6 @ ( one_one @ nat ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
=> ( ! [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Ma2 @ Xa ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_5875_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_p_r_e_d2 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [A6: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ! [Va: nat] :
( ( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
=> ( ! [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_p_r_e_d2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_p_r_e_d_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_5876_VEBT__internal_Oless__shift,axiom,
( ( ord_less @ nat )
= ( ^ [X2: nat,Y4: nat] : ( vEBT_VEBT_less @ ( some @ nat @ X2 ) @ ( some @ nat @ Y4 ) ) ) ) ).
% VEBT_internal.less_shift
thf(fact_5877_is__pred__in__set__def,axiom,
( vEBT_is_pred_in_set
= ( ^ [Xs2: set @ nat,X2: nat,Y4: nat] :
( ( member @ nat @ Y4 @ Xs2 )
& ( ord_less @ nat @ Y4 @ X2 )
& ! [Z7: nat] :
( ( member @ nat @ Z7 @ Xs2 )
=> ( ( ord_less @ nat @ Z7 @ X2 )
=> ( ord_less_eq @ nat @ Z7 @ Y4 ) ) ) ) ) ) ).
% is_pred_in_set_def
thf(fact_5878_VEBT__internal_Opred__none__empty,axiom,
! [Xs: set @ nat,A2: nat] :
( ~ ? [X_12: nat] : ( vEBT_is_pred_in_set @ Xs @ A2 @ X_12 )
=> ( ( finite_finite2 @ nat @ Xs )
=> ~ ? [X6: nat] :
( ( member @ nat @ X6 @ Xs )
& ( ord_less @ nat @ X6 @ A2 ) ) ) ) ).
% VEBT_internal.pred_none_empty
thf(fact_5879_VEBT__internal_Oobtain__set__pred,axiom,
! [Z3: nat,X: nat,A3: set @ nat] :
( ( ord_less @ nat @ Z3 @ X )
=> ( ( vEBT_VEBT_min_in_set @ A3 @ Z3 )
=> ( ( finite_finite2 @ nat @ A3 )
=> ? [X_12: nat] : ( vEBT_is_pred_in_set @ A3 @ X @ X_12 ) ) ) ) ).
% VEBT_internal.obtain_set_pred
thf(fact_5880_VEBT__internal_Opred__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 )
& ! [Z7: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z7 )
& ( ord_less @ nat @ Z7 @ X ) )
=> ( ord_less_eq @ nat @ Z7 @ Y ) ) ) ) ).
% VEBT_internal.pred_member
thf(fact_5881_vebt__inst_Oset__vebt__pred,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_set_vebt @ T2 ) @ X @ Px ) ) ) ).
% vebt_inst.set_vebt_pred
thf(fact_5882_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,B6: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ B6 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) ) ) )
=> ( ( ? [Uv2: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ Uv2 @ Uw ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ N2 ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
thf(fact_5883_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
! [Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
thf(fact_5884_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,B6: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ B6 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Uv2: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ Uv2 @ Uw ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ N2 ) )
=> ( Y
!= ( one_one @ nat ) ) ) )
=> ( ( ? [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
thf(fact_5885_set__vebt__succ,axiom,
! [T2: vEBT_VEBT,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 ) ) ) ).
% set_vebt_succ
thf(fact_5886_vebt__succ_Osimps_I4_J,axiom,
! [V2: product_prod @ nat @ nat,Vc: list @ vEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Vc @ Vd ) @ Ve )
= ( none @ nat ) ) ).
% vebt_succ.simps(4)
thf(fact_5887_vebt__succ_Osimps_I5_J,axiom,
! [V2: product_prod @ nat @ nat,Vg: list @ vEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg @ Vh ) @ Vi )
= ( none @ nat ) ) ).
% vebt_succ.simps(5)
thf(fact_5888_vebt__succ__code_I1_J,axiom,
! [B2: $o,X: nat,A2: $o] :
( ( ( B2
& ( X
= ( zero_zero @ nat ) ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ ( B2
& ( X
= ( zero_zero @ nat ) ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( none @ nat ) ) ) ) ).
% vebt_succ_code(1)
thf(fact_5889_vebt__succ_Osimps_I1_J,axiom,
! [B2: $o,Uu3: $o] :
( ( B2
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu3 @ B2 ) @ ( zero_zero @ nat ) )
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B2
=> ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu3 @ B2 ) @ ( zero_zero @ nat ) )
= ( none @ nat ) ) ) ) ).
% vebt_succ.simps(1)
thf(fact_5890_VEBT__internal_Osucc__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 ) ) ) ) ) ).
% VEBT_internal.succ_empty
thf(fact_5891_VEBT__internal_OgeqmaxNone,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: 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 @ TreeList @ 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 @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) ) ) ).
% VEBT_internal.geqmaxNone
thf(fact_5892_VEBT__internal_Ohelpyd,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 ) ) ) ) ).
% VEBT_internal.helpyd
thf(fact_5893_VEBT__internal_Osucc__min,axiom,
! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( some @ nat @ Mi ) ) ) ) ).
% VEBT_internal.succ_min
thf(fact_5894_VEBT__internal_Osucc__list__to__short,axiom,
! [Deg: nat,Mi: nat,X: nat,TreeList: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) @ ( 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 @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) ) ) ) ).
% VEBT_internal.succ_list_to_short
thf(fact_5895_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
thf(fact_5896_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ B6 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B6 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv2: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ Uv2 @ Uw ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ N2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
=> ( ! [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( one_one @ nat )
@ ( if @ nat @ ( ord_less @ nat @ Xa @ Mi2 ) @ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_m_a_x_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ nat ) )
@ ( if @ nat
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( one_one @ nat )
@ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel2 @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_5897_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: nat] :
( ( ( vEBT_T_s_u_c_c2 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ B6 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B6 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv2: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ Uv2 @ Uw ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ N2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
=> ( ! [Ux: nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Ux @ Uy @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( one_one @ nat ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( if @ nat @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ nat
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( plus_plus @ nat @ ( one_one @ nat ) @ ( vEBT_T_s_u_c_c2 @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( plus_plus @ nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ nat ) ) )
@ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_T_s_u_c_c_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_5898_vebt__pred_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: option @ nat] :
( ( ( vEBT_vebt_pred @ X @ Xa )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
!= ( none @ nat ) ) ) )
=> ( ! [A6: $o] :
( ? [Uw: $o] :
( X
= ( vEBT_Leaf @ A6 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ( ? [Va: nat] :
( Xa
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( B6
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B6
=> ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y
= ( none @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi2 @ Xa ) @ ( 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 @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.elims
thf(fact_5899_is__succ__in__set__def,axiom,
( vEBT_is_succ_in_set
= ( ^ [Xs2: set @ nat,X2: nat,Y4: nat] :
( ( member @ nat @ Y4 @ Xs2 )
& ( ord_less @ nat @ X2 @ Y4 )
& ! [Z7: nat] :
( ( member @ nat @ Z7 @ Xs2 )
=> ( ( ord_less @ nat @ X2 @ Z7 )
=> ( ord_less_eq @ nat @ Y4 @ Z7 ) ) ) ) ) ) ).
% is_succ_in_set_def
thf(fact_5900_VEBT__internal_Osucc__none__empty,axiom,
! [Xs: set @ nat,A2: nat] :
( ~ ? [X_12: nat] : ( vEBT_is_succ_in_set @ Xs @ A2 @ X_12 )
=> ( ( finite_finite2 @ nat @ Xs )
=> ~ ? [X6: nat] :
( ( member @ nat @ X6 @ Xs )
& ( ord_less @ nat @ A2 @ X6 ) ) ) ) ).
% VEBT_internal.succ_none_empty
thf(fact_5901_VEBT__internal_Oobtain__set__succ,axiom,
! [X: nat,Z3: nat,A3: set @ nat,B3: set @ nat] :
( ( ord_less @ nat @ X @ Z3 )
=> ( ( vEBT_VEBT_max_in_set @ A3 @ Z3 )
=> ( ( finite_finite2 @ nat @ B3 )
=> ( ( A3 = B3 )
=> ? [X_12: nat] : ( vEBT_is_succ_in_set @ A3 @ X @ X_12 ) ) ) ) ) ).
% VEBT_internal.obtain_set_succ
thf(fact_5902_VEBT__internal_Osucc__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 )
& ! [Z7: nat] :
( ( ( vEBT_vebt_member @ T2 @ Z7 )
& ( ord_less @ nat @ X @ Z7 ) )
=> ( ord_less_eq @ nat @ Y @ Z7 ) ) ) ) ).
% VEBT_internal.succ_member
thf(fact_5903_vebt__inst_Oset__vebt__succ,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 ) ) ) ).
% vebt_inst.set_vebt_succ
thf(fact_5904_vebt__succ__code_I2_J,axiom,
! [Deg: nat,Info: option @ ( product_prod @ nat @ nat ),TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ( ord_less_eq @ nat @ Deg @ ( one_one @ nat ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( none @ nat ) ) )
& ( ~ ( ord_less_eq @ nat @ Deg @ ( one_one @ nat ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
= ( case_option @ ( option @ nat ) @ ( product_prod @ nat @ nat ) @ ( none @ nat )
@ ( product_case_prod @ nat @ nat @ ( option @ nat )
@ ^ [Mi3: nat,Ma3: nat] :
( if @ ( option @ nat ) @ ( ord_less @ nat @ X @ Mi3 ) @ ( some @ nat @ Mi3 )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ 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 @ TreeList @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) )
@ Info ) ) ) ) ).
% vebt_succ_code(2)
thf(fact_5905_vebt__succ_Osimps_I6_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: 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 @ Va2 ) ) @ TreeList @ 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 @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_succ.simps(6)
thf(fact_5906_vebt__pred_Osimps_I7_J,axiom,
! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList: 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 @ Va2 ) ) @ TreeList @ 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 @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( some @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% vebt_pred.simps(7)
thf(fact_5907_VEBT__internal_Osucc__less__length__list,axiom,
! [Deg: nat,Mi: nat,X: nat,TreeList: list @ vEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.succ_less_length_list
thf(fact_5908_VEBT__internal_Osucc__greatereq__min,axiom,
! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ Mi @ X )
=> ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ 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 ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% VEBT_internal.succ_greatereq_min
thf(fact_5909_VEBT__internal_Opred__less__length__list,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X @ Ma )
=> ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.pred_less_length_list
thf(fact_5910_VEBT__internal_Opred__lesseq__max,axiom,
! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( ord_less_eq @ nat @ X @ Ma )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ 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 ) @ TreeList ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( 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 @ TreeList @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ).
% VEBT_internal.pred_lesseq_max
thf(fact_5911_vebt__succ_Oelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: option @ nat] :
( ( ( vEBT_vebt_succ @ X @ Xa )
= Y )
=> ( ! [Uu2: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ B6 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ~ ( ( B6
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B6
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ( ( ? [Uv2: $o,Uw: $o] :
( X
= ( vEBT_Leaf @ Uv2 @ Uw ) )
=> ( ? [N2: nat] :
( Xa
= ( suc @ N2 ) )
=> ( 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 ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ( ( ? [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( Y
!= ( none @ nat ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
@ ( none @ nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.elims
thf(fact_5912_vebt__succ_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: option @ nat] :
( ( ( vEBT_vebt_succ @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ B6 ) )
=> ( ( Xa
= ( zero_zero @ nat ) )
=> ( ( ( B6
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B6
=> ( Y
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ B6 ) @ ( zero_zero @ nat ) ) ) ) ) )
=> ( ! [Uv2: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ Uv2 @ Uw ) )
=> ! [N2: nat] :
( ( Xa
= ( suc @ N2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uv2 @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
=> ( ! [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 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vc2: list @ vEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vc2 @ Vd2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vg2: list @ vEBT_VEBT,Vh2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_succ_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( some @ nat @ Mi2 ) ) )
& ( ~ ( ord_less @ nat @ Xa @ Mi2 )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_less @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.pelims
thf(fact_5913_vebt__pred_Opelims,axiom,
! [X: vEBT_VEBT,Xa: nat,Y: option @ nat] :
( ( ( vEBT_vebt_pred @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Xa
= ( 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 ) ) ) ) ) )
=> ( ! [A6: $o,Uw: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ Uw ) )
=> ( ( Xa
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y
= ( none @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ Uw ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
=> ( ! [A6: $o,B6: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B6 ) )
=> ! [Va: nat] :
( ( Xa
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( B6
=> ( Y
= ( some @ nat @ ( one_one @ nat ) ) ) )
& ( ~ B6
=> ( ( A6
=> ( Y
= ( some @ nat @ ( zero_zero @ nat ) ) ) )
& ( ~ A6
=> ( Y
= ( none @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B6 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
=> ( ! [Uy: nat,Uz2: list @ vEBT_VEBT,Va3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uy @ Uz2 @ Va3 ) )
=> ( ( 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 @ Va3 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vd2: list @ vEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Vd2 @ Ve2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vh2: list @ vEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) )
=> ( ( Y
= ( none @ nat ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_pred_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( ( ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( some @ nat @ Ma2 ) ) )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( Y
= ( if @ ( option @ nat ) @ ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
@ ( if @ ( option @ nat )
@ ( ( ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
!= ( none @ nat ) )
& ( vEBT_VEBT_greater @ ( some @ nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( if @ ( option @ nat )
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( none @ nat ) )
@ ( if @ ( option @ nat ) @ ( ord_less @ nat @ Mi2 @ Xa ) @ ( 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 @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth @ vEBT_VEBT @ TreeList2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( 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 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.pelims
thf(fact_5914_take__bit__numeral__minus__numeral__int,axiom,
! [M: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int )
@ ^ [Q6: num] : ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ int @ Q6 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_5915_take__bit__num__simps_I1_J,axiom,
! [M: num] :
( ( bit_take_bit_num @ ( zero_zero @ nat ) @ M )
= ( none @ num ) ) ).
% take_bit_num_simps(1)
thf(fact_5916_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_5917_take__bit__num__simps_I5_J,axiom,
! [R3: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R3 ) @ one2 )
= ( some @ num @ one2 ) ) ).
% take_bit_num_simps(5)
thf(fact_5918_take__bit__num__simps_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q6: num] : ( some @ num @ ( bit0 @ Q6 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_5919_take__bit__num__simps_I4_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_5920_take__bit__num__simps_I6_J,axiom,
! [R3: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R3 ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q6: num] : ( some @ num @ ( bit0 @ Q6 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_5921_take__bit__num__simps_I7_J,axiom,
! [R3: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R3 ) @ ( bit1 @ M ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ) ).
% take_bit_num_simps(7)
thf(fact_5922_take__bit__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ) ).
% take_bit_numeral_numeral
thf(fact_5923_take__bit__num__eq__Some__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: num,Q3: num] :
( ( ( bit_take_bit_num @ M @ N )
= ( some @ num @ Q3 ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ Q3 ) ) ) ) ).
% take_bit_num_eq_Some_imp
thf(fact_5924_take__bit__num__eq__None__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: num] :
( ( ( bit_take_bit_num @ M @ N )
= ( none @ num ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_num_eq_None_imp
thf(fact_5925_and__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_5926_and__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_5927_and__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_5928_and__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_5929_and__not__num_Osimps_I1_J,axiom,
( ( bit_and_not_num @ one2 @ one2 )
= ( none @ num ) ) ).
% and_not_num.simps(1)
thf(fact_5930_and__not__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_and_not_num @ one2 @ ( bit0 @ N ) )
= ( some @ num @ one2 ) ) ).
% and_not_num.simps(2)
thf(fact_5931_and__not__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ one2 )
= ( some @ num @ ( bit0 @ M ) ) ) ).
% and_not_num.simps(4)
thf(fact_5932_and__not__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_and_not_num @ one2 @ ( bit1 @ N ) )
= ( none @ num ) ) ).
% and_not_num.simps(3)
thf(fact_5933_and__not__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ one2 )
= ( some @ num @ ( bit0 @ M ) ) ) ).
% and_not_num.simps(7)
thf(fact_5934_and__not__num__eq__Some__iff,axiom,
! [M: num,N: num,Q3: num] :
( ( ( bit_and_not_num @ M @ N )
= ( some @ num @ Q3 ) )
= ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( numeral_numeral @ int @ Q3 ) ) ) ).
% and_not_num_eq_Some_iff
thf(fact_5935_and__not__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(8)
thf(fact_5936_and__not__num__eq__None__iff,axiom,
! [M: num,N: num] :
( ( ( bit_and_not_num @ M @ N )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( zero_zero @ int ) ) ) ).
% and_not_num_eq_None_iff
thf(fact_5937_int__numeral__not__and__num,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ N @ M ) ) ) ).
% int_numeral_not_and_num
thf(fact_5938_int__numeral__and__not__num,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ N ) ) ) ).
% int_numeral_and_not_num
thf(fact_5939_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_5940_test__def,axiom,
( vEBT_Intf_test
= ( ^ [N3: nat,Xs2: list @ nat] :
( map @ nat @ nat
@ ^ [X2: nat] : ( if @ nat @ ( vEBT_vebt_member @ ( foldl @ vEBT_VEBT @ nat @ vEBT_vebt_insert @ ( vEBT_vebt_buildup @ N3 ) @ ( cons @ nat @ ( zero_zero @ nat ) @ Xs2 ) ) @ X2 ) @ X2 @ ( the2 @ nat @ ( vEBT_vebt_pred @ ( foldl @ vEBT_VEBT @ nat @ vEBT_vebt_insert @ ( vEBT_vebt_buildup @ N3 ) @ ( cons @ nat @ ( zero_zero @ nat ) @ Xs2 ) ) @ X2 ) ) ) ) ) ) ).
% test_def
thf(fact_5941_set__vebt__pred_H,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( ? [X6: nat] :
( ( member @ nat @ X6 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X6 @ X ) )
=> ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat
@ ( order_Greatest @ nat
@ ^ [Y4: nat] :
( ( member @ nat @ Y4 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ Y4 @ X ) ) ) ) ) )
& ( ~ ? [X3: nat] :
( ( member @ nat @ X3 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X3 @ X ) )
=> ( ( vEBT_vebt_pred @ T2 @ X )
= ( none @ nat ) ) ) ) ) ).
% set_vebt_pred'
thf(fact_5942_list_Oinject,axiom,
! [A: $tType,X21: A,X222: list @ A,Y21: A,Y222: list @ A] :
( ( ( cons @ A @ X21 @ X222 )
= ( cons @ A @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_5943_or_Oidem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ A2 )
= A2 ) ) ).
% or.idem
thf(fact_5944_or_Oleft__idem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ).
% or.left_idem
thf(fact_5945_or_Oright__idem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) @ B2 )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ).
% or.right_idem
thf(fact_5946_or_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% or.left_neutral
thf(fact_5947_or_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% or.right_neutral
thf(fact_5948_bex__empty,axiom,
! [A: $tType,P: A > $o] :
~ ? [X6: A] :
( ( member @ A @ X6 @ ( bot_bot @ ( set @ A ) ) )
& ( P @ X6 ) ) ).
% bex_empty
thf(fact_5949_take__bit__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) ) ) ).
% take_bit_or
thf(fact_5950_push__bit__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_or
thf(fact_5951_finite__Collect__bex,axiom,
! [B: $tType,A: $tType,A3: set @ A,Q: B > A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
? [Y4: A] :
( ( member @ A @ Y4 @ A3 )
& ( Q @ X2 @ Y4 ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Y4: B] : ( Q @ Y4 @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_5952_nth__Cons__Suc,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ N ) )
= ( nth @ A @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_5953_nth__Cons__0,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) )
= X ) ).
% nth_Cons_0
thf(fact_5954_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( set2 @ A @ ( cons @ A @ X21 @ X222 ) )
= ( insert @ A @ X21 @ ( set2 @ A @ X222 ) ) ) ).
% list.simps(15)
thf(fact_5955_bit_Odisj__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_left
thf(fact_5956_bit_Odisj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_right
thf(fact_5957_or__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% or_nonnegative_int_iff
thf(fact_5958_or__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
| ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% or_negative_int_iff
thf(fact_5959_bit_Ode__Morgan__conj,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344872417868541ns_and @ A @ X @ Y ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ ( bit_ri4277139882892585799ns_not @ A @ Y ) ) ) ) ).
% bit.de_Morgan_conj
thf(fact_5960_bit_Ode__Morgan__disj,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_se1065995026697491101ons_or @ A @ X @ Y ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ ( bit_ri4277139882892585799ns_not @ A @ Y ) ) ) ) ).
% bit.de_Morgan_disj
thf(fact_5961_or__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% or_numerals(8)
thf(fact_5962_or__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(2)
thf(fact_5963_bit_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_right
thf(fact_5964_bit_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_left
thf(fact_5965_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_5966_or__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% or_numerals(5)
thf(fact_5967_or__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(1)
thf(fact_5968_nth__Cons__numeral,axiom,
! [A: $tType,X: A,Xs: list @ A,V2: num] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( numeral_numeral @ nat @ V2 ) )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_5969_or__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_5970_or__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_5971_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_5972_or__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_5973_and__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se1065995026697491101ons_or @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_5974_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_5975_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_5976_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_5977_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list @ nat,Q3: nat] :
( ( ( cons @ nat @ M @ ( cons @ nat @ N @ Ns ) )
= ( upt @ M @ Q3 ) )
= ( ( cons @ nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q3 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_5978_list__update__code_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I: nat,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons @ A @ X @ ( list_update @ A @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_5979_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y4: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y4 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_5980_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y4: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y4 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_5981_plus__and__or,axiom,
! [X: int,Y: int] :
( ( plus_plus @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) )
= ( plus_plus @ int @ X @ Y ) ) ).
% plus_and_or
thf(fact_5982_or__eq__not__not__and,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se1065995026697491101ons_or @ A )
= ( ^ [A5: A,B5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ A5 ) @ ( bit_ri4277139882892585799ns_not @ A @ B5 ) ) ) ) ) ) ).
% or_eq_not_not_and
thf(fact_5983_and__eq__not__not__or,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344872417868541ns_and @ A )
= ( ^ [A5: A,B5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ A5 ) @ ( bit_ri4277139882892585799ns_not @ A @ B5 ) ) ) ) ) ) ).
% and_eq_not_not_or
thf(fact_5984_remove1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( remove1 @ A @ X @ ( cons @ A @ Y @ Xs ) )
= Xs ) )
& ( ( X != Y )
=> ( ( remove1 @ A @ X @ ( cons @ A @ Y @ Xs ) )
= ( cons @ A @ Y @ ( remove1 @ A @ X @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_5985_of__int__or__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: int,L: int] :
( ( ring_1_of_int @ A @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) )
= ( bit_se1065995026697491101ons_or @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) ) ) ).
% of_int_or_eq
thf(fact_5986_bit_Oconj__disj__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_se1065995026697491101ons_or @ A @ Y @ Z3 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ X @ Y ) @ ( bit_se5824344872417868541ns_and @ A @ X @ Z3 ) ) ) ) ).
% bit.conj_disj_distrib
thf(fact_5987_bit_Odisj__conj__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( bit_se5824344872417868541ns_and @ A @ Y @ Z3 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ X @ Y ) @ ( bit_se1065995026697491101ons_or @ A @ X @ Z3 ) ) ) ) ).
% bit.disj_conj_distrib
thf(fact_5988_bit_Oconj__disj__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y: A,Z3: A,X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ Y @ Z3 ) @ X )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ Y @ X ) @ ( bit_se5824344872417868541ns_and @ A @ Z3 @ X ) ) ) ) ).
% bit.conj_disj_distrib2
thf(fact_5989_bit_Odisj__conj__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y: A,Z3: A,X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ Y @ Z3 ) @ X )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ Y @ X ) @ ( bit_se1065995026697491101ons_or @ A @ Z3 @ X ) ) ) ) ).
% bit.disj_conj_distrib2
thf(fact_5990_Bex__def,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A4: set @ A,P3: A > $o] :
? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( P3 @ X2 ) ) ) ) ).
% Bex_def
thf(fact_5991_foldl__Cons,axiom,
! [B: $tType,A: $tType,F2: B > A > B,A2: B,X: A,Xs: list @ A] :
( ( foldl @ B @ A @ F2 @ A2 @ ( cons @ A @ X @ Xs ) )
= ( foldl @ B @ A @ F2 @ ( F2 @ A2 @ X ) @ Xs ) ) ).
% foldl_Cons
thf(fact_5992_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_5993_or_Oassoc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) @ C2 )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ ( bit_se1065995026697491101ons_or @ A @ B2 @ C2 ) ) ) ) ).
% or.assoc
thf(fact_5994_or_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se1065995026697491101ons_or @ A )
= ( ^ [A5: A,B5: A] : ( bit_se1065995026697491101ons_or @ A @ B5 @ A5 ) ) ) ) ).
% or.commute
thf(fact_5995_or_Oleft__commute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( bit_se1065995026697491101ons_or @ A @ B2 @ ( bit_se1065995026697491101ons_or @ A @ A2 @ C2 ) )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ ( bit_se1065995026697491101ons_or @ A @ B2 @ C2 ) ) ) ) ).
% or.left_commute
thf(fact_5996_bit__or__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ).
% bit_or_iff
thf(fact_5997_bit__or__int__iff,axiom,
! [K: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
| ( bit_se5641148757651400278ts_bit @ int @ L @ N ) ) ) ).
% bit_or_int_iff
thf(fact_5998_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_5999_or__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% or_eq_0_iff
thf(fact_6000_of__nat__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se1065995026697491101ons_or @ nat @ M @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_or_eq
thf(fact_6001_Cons__in__shuffles__rightI,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A,Z3: A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ Z3 @ Zs ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Z3 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_6002_Cons__in__shuffles__leftI,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A,Z3: A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ Z3 @ Zs ) @ ( shuffles @ A @ ( cons @ A @ Z3 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_6003_distinct__length__2__or__more,axiom,
! [A: $tType,A2: A,B2: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ A2 @ ( cons @ A @ B2 @ Xs ) ) )
= ( ( A2 != B2 )
& ( distinct @ A @ ( cons @ A @ A2 @ Xs ) )
& ( distinct @ A @ ( cons @ A @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_6004_foldl__cong,axiom,
! [A: $tType,B: $tType,A2: A,B2: A,L: list @ B,K: list @ B,F2: A > B > A,G: A > B > A] :
( ( A2 = B2 )
=> ( ( L = K )
=> ( ! [A6: A,X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ L ) )
=> ( ( F2 @ A6 @ X3 )
= ( G @ A6 @ X3 ) ) )
=> ( ( foldl @ A @ B @ F2 @ A2 @ L )
= ( foldl @ A @ B @ G @ B2 @ K ) ) ) ) ) ).
% foldl_cong
thf(fact_6005_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y: A,X222: list @ A,X21: A] :
( ( member @ A @ Y @ ( set2 @ A @ X222 ) )
=> ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_6006_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] : ( member @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_6007_list_Oset__cases,axiom,
! [A: $tType,E2: A,A2: list @ A] :
( ( member @ A @ E2 @ ( set2 @ A @ A2 ) )
=> ( ! [Z23: list @ A] :
( A2
!= ( cons @ A @ E2 @ Z23 ) )
=> ~ ! [Z12: A,Z23: list @ A] :
( ( A2
= ( cons @ A @ Z12 @ Z23 ) )
=> ~ ( member @ A @ E2 @ ( set2 @ A @ Z23 ) ) ) ) ) ).
% list.set_cases
thf(fact_6008_set__ConsD,axiom,
! [A: $tType,Y: A,X: A,Xs: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_6009_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F2: A > B,X21: A,X222: list @ A] :
( ( map @ A @ B @ F2 @ ( cons @ A @ X21 @ X222 ) )
= ( cons @ B @ ( F2 @ X21 ) @ ( map @ A @ B @ F2 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_6010_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F2: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
=> ? [Z: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map @ B @ A @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_6011_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F2: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ? [Z: B,Zs2: list @ B] :
( ( Xs
= ( cons @ B @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y )
& ( ( map @ B @ A @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_6012_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F2: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
= ( ? [Z7: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z7 @ Zs3 ) )
& ( X
= ( F2 @ Z7 ) )
& ( Xs
= ( map @ B @ A @ F2 @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_6013_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ? [Z7: B,Zs3: list @ B] :
( ( Xs
= ( cons @ B @ Z7 @ Zs3 ) )
& ( ( F2 @ Z7 )
= Y )
& ( ( map @ B @ A @ F2 @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_6014_disjunctive__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ! [N2: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) )
=> ( ( plus_plus @ A @ A2 @ B2 )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ) ).
% disjunctive_add
thf(fact_6015_OR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) ) ) ) ).
% OR_lower
thf(fact_6016_or__greater__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ K @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) ) ) ).
% or_greater_eq
thf(fact_6017_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs
!= ( cons @ A @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_6018_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_6019_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [F3: A > B,A4: set @ A] :
( collect @ B
@ ^ [Y4: B] :
? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( Y4
= ( F3 @ X2 ) ) ) ) ) ) ).
% image_def
thf(fact_6020_Cons__shuffles__subset2,axiom,
! [A: $tType,Y: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_6021_Cons__shuffles__subset1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_6022_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_6023_or__int__def,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L2: int] : ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ K3 ) @ ( bit_ri4277139882892585799ns_not @ int @ L2 ) ) ) ) ) ).
% or_int_def
thf(fact_6024_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_6025_list__update__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) @ Y )
= ( cons @ A @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_6026_distinct_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_6027_shuffles_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_6028_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,G: A > B > A,A2: A,F2: C > B,Xs: list @ C] :
( ( foldl @ A @ B @ G @ A2 @ ( map @ C @ B @ F2 @ Xs ) )
= ( foldl @ A @ C
@ ^ [A5: A,X2: C] : ( G @ A5 @ ( F2 @ X2 ) )
@ A2
@ Xs ) ) ).
% foldl_map
thf(fact_6029_find_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( some @ A @ X ) ) )
& ( ~ ( P @ X )
=> ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( find @ A @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_6030_Cons__in__subseqsD,axiom,
! [A: $tType,Y: A,Ys: list @ A,Xs: list @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ Y @ Ys ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_6031_or__not__numerals_I1_J,axiom,
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% or_not_numerals(1)
thf(fact_6032_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X2: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_6033_bit_Oxor__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [X2: A,Y4: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ Y4 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ Y4 ) ) ) ) ) ).
% bit.xor_def
thf(fact_6034_bit_Oxor__def2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [X2: A,Y4: A] : ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ X2 @ Y4 ) @ ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ ( bit_ri4277139882892585799ns_not @ A @ Y4 ) ) ) ) ) ) ).
% bit.xor_def2
thf(fact_6035_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 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X @ X2 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) ) ) ) ).
% sorted_simps(2)
thf(fact_6036_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X @ Ys ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
=> ( ord_less @ A @ X @ X2 ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_6037_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N3: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_6038_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons @ nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_6039_xor__int__def,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L2: int] : ( bit_se1065995026697491101ons_or @ int @ ( bit_se5824344872417868541ns_and @ int @ K3 @ ( bit_ri4277139882892585799ns_not @ int @ L2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ K3 ) @ L2 ) ) ) ) ).
% xor_int_def
thf(fact_6040_concat__bit__def,axiom,
( bit_concat_bit
= ( ^ [N3: nat,K3: int,L2: int] : ( bit_se1065995026697491101ons_or @ int @ ( bit_se2584673776208193580ke_bit @ int @ N3 @ K3 ) @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ L2 ) ) ) ) ).
% concat_bit_def
thf(fact_6041_set__bit__int__def,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N3: nat,K3: int] : ( bit_se1065995026697491101ons_or @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ).
% set_bit_int_def
thf(fact_6042_count__list_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y ) @ ( one_one @ nat ) ) ) )
& ( ( X != Y )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( count_list @ A @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_6043_even__or__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_or_iff
thf(fact_6044_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_6045_or__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(2)
thf(fact_6046_or__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) ) ).
% or_not_numerals(4)
thf(fact_6047_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X222 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size(4)
thf(fact_6048_nth__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_6049_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X @ Xs ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_6050_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X21: A,X222: list @ A] :
( ( size_list @ A @ X @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_6051_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_6052_or__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(3)
thf(fact_6053_map__upt__Suc,axiom,
! [A: $tType,F2: nat > A,N: nat] :
( ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( cons @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I4: nat] : ( F2 @ ( suc @ I4 ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_6054_or__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% or_not_numerals(7)
thf(fact_6055_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A,N: nat] :
( ( X != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= Y )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_6056_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_6057_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat,Y: A] :
( ( ( cons @ A @ X @ Xs )
= ( replicate @ A @ N @ Y ) )
= ( ( X = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_6058_signed__take__bit__eq__if__negative,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
=> ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ).
% signed_take_bit_eq_if_negative
thf(fact_6059_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_6060_or__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% or_not_numerals(6)
thf(fact_6061_Pow__set_I2_J,axiom,
! [B: $tType,X: B,Xs: list @ B] :
( ( pow2 @ B @ ( set2 @ B @ ( cons @ B @ X @ Xs ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) @ ( image @ ( set @ B ) @ ( set @ B ) @ ( insert @ B @ X ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_6062_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( one_one @ A ) )
= ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% or_one_eq
thf(fact_6063_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A2 )
= ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% one_or_eq
thf(fact_6064_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_6065_OR__upper,axiom,
! [X: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_6066_or__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( 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 @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_6067_signed__take__bit__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A5 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ N3 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ) ) ) ).
% signed_take_bit_def
thf(fact_6068_or__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( 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 @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_6069_or__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( 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 @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_6070_vebt__inst_Oset__vebt__pred_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ? [X6: nat] :
( ( member @ nat @ X6 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X6 @ X ) )
=> ( ( vEBT_vebt_pred @ T2 @ X )
= ( some @ nat
@ ( order_Greatest @ nat
@ ^ [Y4: nat] :
( ( member @ nat @ Y4 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ Y4 @ X ) ) ) ) ) )
& ( ~ ? [X3: nat] :
( ( member @ nat @ X3 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X3 @ X ) )
=> ( ( vEBT_vebt_pred @ T2 @ X )
= ( none @ nat ) ) ) ) ) ).
% vebt_inst.set_vebt_pred'
thf(fact_6071_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_6072_map__project__def,axiom,
! [B: $tType,A: $tType] :
( ( map_project @ A @ B )
= ( ^ [F3: A > ( option @ B ),A4: set @ A] :
( collect @ B
@ ^ [B5: B] :
? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ( F3 @ X2 )
= ( some @ B @ B5 ) ) ) ) ) ) ).
% map_project_def
thf(fact_6073_set__vebt__succ_H,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_invar_vebt @ T2 @ n )
=> ( ( ? [X6: nat] :
( ( member @ nat @ X6 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X @ X6 ) )
=> ( ( vEBT_vebt_succ @ T2 @ X )
= ( some @ nat
@ ( ord_Least @ nat
@ ^ [Y4: nat] :
( ( member @ nat @ Y4 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X @ Y4 ) ) ) ) ) )
& ( ~ ? [X3: nat] :
( ( member @ nat @ X3 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X @ X3 ) )
=> ( ( vEBT_vebt_succ @ T2 @ X )
= ( none @ nat ) ) ) ) ) ).
% set_vebt_succ'
thf(fact_6074_or__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_6075_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_6076_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_6077_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_6078_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_6079_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_6080_or__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_6081_or__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_6082_or__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_6083_or__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_6084_or__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_6085_set__bit__nat__def,axiom,
( ( bit_se5668285175392031749et_bit @ nat )
= ( ^ [M3: nat,N3: nat] : ( bit_se1065995026697491101ons_or @ nat @ N3 @ ( bit_se4730199178511100633sh_bit @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ).
% set_bit_nat_def
thf(fact_6086_not__less__Least,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [K: A,P: A > $o] :
( ( ord_less @ A @ K @ ( ord_Least @ A @ P ) )
=> ~ ( P @ K ) ) ) ).
% not_less_Least
thf(fact_6087_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_6088_Least__le,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ K ) ) ) ).
% Least_le
thf(fact_6089_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X6 @ Y3 ) )
& ! [Y3: A] :
( ( ( P @ Y3 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y3 @ Ya2 ) ) )
=> ( Y3 = X6 ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_6090_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,Z3: A] :
( ? [X6: A] :
( ( P @ X6 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X6 @ Y3 ) )
& ! [Y3: A] :
( ( ( P @ Y3 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y3 @ Ya2 ) ) )
=> ( Y3 = X6 ) ) )
=> ( ( P @ Z3 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z3 ) ) ) ) ).
% Least1_le
thf(fact_6091_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_6092_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( ord_Least @ A @ P )
= X ) ) ) ) ).
% Least_equality
thf(fact_6093_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B12: A] :
( ( P @ B12 )
=> ( ord_less_eq @ A @ A6 @ B12 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_6094_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B12: A] :
( ( P @ B12 )
=> ( ord_less_eq @ A @ A6 @ B12 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_6095_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one2 @ one2 )
= one2 ) ).
% or_not_num_neg.simps(1)
thf(fact_6096_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_6097_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI_ex
thf(fact_6098_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_6099_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
( ( P @ N )
=> ( ( Q @ M )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ! [K2: nat] :
( ( P @ ( suc @ K2 ) )
= ( Q @ K2 ) )
=> ( ( ord_Least @ nat @ P )
= ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_6100_or__not__num__neg_Osimps_I4_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one2 )
= ( bit0 @ one2 ) ) ).
% or_not_num_neg.simps(4)
thf(fact_6101_or__not__num__neg_Osimps_I6_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
= ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(6)
thf(fact_6102_or__not__num__neg_Osimps_I3_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one2 @ ( bit1 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(3)
thf(fact_6103_or__not__num__neg_Osimps_I7_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one2 )
= one2 ) ).
% or_not_num_neg.simps(7)
thf(fact_6104_or__not__num__neg_Osimps_I5_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(5)
thf(fact_6105_or__not__num__neg_Osimps_I9_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(9)
thf(fact_6106_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( suc
@ ( ord_Least @ nat
@ ^ [M3: nat] : ( P @ ( suc @ M3 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_6107_or__nat__def,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% or_nat_def
thf(fact_6108_Least__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ? [X_1: A] : ( P @ X_1 )
=> ( ( ord_Least @ A @ P )
= ( lattic643756798350308766er_Min @ A @ ( collect @ A @ P ) ) ) ) ) ) ).
% Least_Min
thf(fact_6109_or__not__num__neg_Osimps_I2_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one2 @ ( bit0 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(2)
thf(fact_6110_or__not__num__neg_Osimps_I8_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(8)
thf(fact_6111_or__not__num__neg_Oelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( ( bit_or_not_num_neg @ X @ Xa )
= Y )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y != one2 ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y
!= ( bit1 @ M4 ) ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y
!= ( bit1 @ M4 ) ) ) )
=> ( ( ? [N2: num] :
( X
= ( bit0 @ N2 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( bit0 @ one2 ) ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit0 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit0 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y
!= ( bit0 @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
=> ( ( ? [N2: num] :
( X
= ( bit1 @ N2 ) )
=> ( ( Xa = one2 )
=> ( Y != one2 ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit1 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
=> ~ ! [N2: num] :
( ( X
= ( bit1 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y
!= ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
thf(fact_6112_int__numeral__or__not__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% int_numeral_or_not_num_neg
thf(fact_6113_int__numeral__not__or__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% int_numeral_not_or_num_neg
thf(fact_6114_numeral__or__not__num__eq,axiom,
! [M: num,N: num] :
( ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N ) )
= ( uminus_uminus @ int @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% numeral_or_not_num_eq
thf(fact_6115_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_6116_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_6117_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M3: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_6118_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M3: nat,N3: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ N3
@ ( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ M3
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_6119_vebt__inst_Oset__vebt__succ_H,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ? [X6: nat] :
( ( member @ nat @ X6 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X @ X6 ) )
=> ( ( vEBT_vebt_succ @ T2 @ X )
= ( some @ nat
@ ( ord_Least @ nat
@ ^ [Y4: nat] :
( ( member @ nat @ Y4 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X @ Y4 ) ) ) ) ) )
& ( ~ ? [X3: nat] :
( ( member @ nat @ X3 @ ( vEBT_set_vebt @ T2 ) )
& ( ord_less @ nat @ X @ X3 ) )
=> ( ( vEBT_vebt_succ @ T2 @ X )
= ( none @ nat ) ) ) ) ) ).
% vebt_inst.set_vebt_succ'
thf(fact_6120_or__not__num__neg_Opelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( ( bit_or_not_num_neg @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( ( Y
= ( bit1 @ M4 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ M4 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( ( Y
= ( bit1 @ M4 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ M4 ) ) ) ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit0 @ N2 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( bit0 @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N2 ) @ one2 ) ) ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit0 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N2 ) @ ( bit0 @ M4 ) ) ) ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit0 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( ( Y
= ( bit0 @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N2 ) @ ( bit1 @ M4 ) ) ) ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit1 @ N2 ) )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N2 ) @ one2 ) ) ) ) )
=> ( ! [N2: num] :
( ( X
= ( bit1 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N2 ) @ ( bit0 @ M4 ) ) ) ) ) )
=> ~ ! [N2: num] :
( ( X
= ( bit1 @ N2 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N2 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
thf(fact_6121_arg__min__SOME__Min,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S2: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( lattic7623131987881927897min_on @ A @ B @ F2 @ S2 )
= ( fChoice @ A
@ ^ [Y4: A] :
( ( member @ A @ Y4 @ S2 )
& ( ( F2 @ Y4 )
= ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F2 @ S2 ) ) ) ) ) ) ) ) ).
% arg_min_SOME_Min
thf(fact_6122_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S2: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S2 ) @ S2 ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_6123_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S2: set @ A,Y: A,F2: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y @ S2 )
=> ( ord_less_eq @ B @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S2 ) ) @ ( F2 @ Y ) ) ) ) ) ) ).
% arg_min_least
thf(fact_6124_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S2: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X6: A] :
( ( member @ A @ X6 @ S2 )
& ( ord_less @ B @ ( F2 @ X6 ) @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S2 ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_6125_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N3: nat] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N3 @ M ) ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_6126_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_6127_case__nat__numeral,axiom,
! [A: $tType,A2: A,F2: nat > A,V2: num] :
( ( case_nat @ A @ A2 @ F2 @ ( numeral_numeral @ nat @ V2 ) )
= ( F2 @ ( pred_numeral @ V2 ) ) ) ).
% case_nat_numeral
thf(fact_6128_case__nat__add__eq__if,axiom,
! [A: $tType,A2: A,F2: nat > A,V2: num,N: nat] :
( ( case_nat @ A @ A2 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_6129_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H2: A > B,F1: A,F22: nat > A,Nat: nat] :
( ( H2 @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( case_nat @ B @ ( H2 @ F1 )
@ ^ [X2: nat] : ( H2 @ ( F22 @ X2 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_6130_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_6131_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: nat > A,X22: nat] :
( ( case_nat @ A @ F1 @ F22 @ ( suc @ X22 ) )
= ( F22 @ X22 ) ) ).
% old.nat.simps(5)
thf(fact_6132_nth__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( case_nat @ A @ X @ ( nth @ A @ Xs ) @ N ) ) ).
% nth_Cons
thf(fact_6133_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X2: A,F3: nat > A,N3: nat] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ X2
@ ( F3 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_6134_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ N @ one2 )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N3: nat] : ( some @ num @ one2 )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_6135_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N3: nat] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q6: num] : ( some @ num @ ( bit0 @ Q6 ) )
@ ( bit_take_bit_num @ N3 @ M ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_6136_Bit__Operations_Otake__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N3: nat,M3: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A5: nat,X2: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one2 )
@ ^ [P6: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q6: num] : ( some @ num @ ( bit0 @ Q6 ) )
@ ( bit_take_bit_num @ O @ P6 ) )
@ ^ [P6: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
@ X2 )
@ A5 )
@ ( product_Pair @ nat @ num @ N3 @ M3 ) ) ) ) ).
% Bit_Operations.take_bit_num_code
thf(fact_6137_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > $o,X: option @ A,R2: B > $o,F2: B,G: A > B] :
( ( case_option @ $o @ A @ P @ Q @ X )
=> ( ( P
=> ( R2 @ F2 ) )
=> ( ! [Q5: A] :
( ( Q @ Q5 )
=> ( R2 @ ( G @ Q5 ) ) )
=> ( R2 @ ( case_option @ B @ A @ F2 @ G @ X ) ) ) ) ) ).
% disjE_realizer2
thf(fact_6138_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat
= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $true
@ ^ [Uu: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_6139_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat
!= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $false
@ ^ [Uu: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_6140_verit__eq__simplify_I17_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A,X22: num] :
( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X22 ) )
= ( F22 @ X22 ) ) ).
% verit_eq_simplify(17)
thf(fact_6141_verit__eq__simplify_I16_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A] :
( ( case_num @ A @ F1 @ F22 @ F32 @ one2 )
= F1 ) ).
% verit_eq_simplify(16)
thf(fact_6142_num_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H2: A > B,F1: A,F22: num > A,F32: num > A,Num: num] :
( ( H2 @ ( case_num @ A @ F1 @ F22 @ F32 @ Num ) )
= ( case_num @ B @ ( H2 @ F1 )
@ ^ [X2: num] : ( H2 @ ( F22 @ X2 ) )
@ ^ [X2: num] : ( H2 @ ( F32 @ X2 ) )
@ Num ) ) ).
% num.case_distrib
thf(fact_6143_verit__eq__simplify_I18_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A,X32: num] :
( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit1 @ X32 ) )
= ( F32 @ X32 ) ) ).
% verit_eq_simplify(18)
thf(fact_6144_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
= ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_6145_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M5: nat] : ( suc @ ( ord_max @ nat @ M5 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_6146_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M5: nat] : ( suc @ ( ord_max @ nat @ N @ M5 ) )
@ M ) ) ).
% max_Suc1
thf(fact_6147_list__update_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I: nat,V2: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ I @ V2 )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ V2 @ Xs )
@ ^ [J3: nat] : ( cons @ A @ X @ ( list_update @ A @ Xs @ J3 @ V2 ) )
@ I ) ) ).
% list_update.simps(2)
thf(fact_6148_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K3: nat] : K3
@ ( minus_minus @ nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_6149_bit__numeral__rec_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W2 ) ) @ N )
= ( case_nat @ $o @ $false @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W2 ) ) @ N ) ) ) ).
% bit_numeral_rec(1)
thf(fact_6150_bit__numeral__rec_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W2 ) ) @ N )
= ( case_nat @ $o @ $true @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W2 ) ) @ N ) ) ) ).
% bit_numeral_rec(2)
thf(fact_6151_Pow__fold,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( pow2 @ A @ A3 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A,A4: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A4 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X2 ) @ A4 ) )
@ ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A3 ) ) ) ).
% Pow_fold
thf(fact_6152_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) )
=> ( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ) ).
% shuffles.psimps(3)
thf(fact_6153_union__fold__insert,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
= ( finite_fold @ A @ ( set @ A ) @ ( insert @ A ) @ B3 @ A3 ) ) ) ).
% union_fold_insert
thf(fact_6154_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A3 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X @ A3 ) ) ) ) ).
% Max.eq_fold
thf(fact_6155_image__fold__insert,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( image @ A @ B @ F2 @ A3 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K3: A] : ( insert @ B @ ( F2 @ K3 ) )
@ ( bot_bot @ ( set @ B ) )
@ A3 ) ) ) ).
% image_fold_insert
thf(fact_6156_shuffles_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( insert @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( insert @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X3: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs3 ) )
=> ! [Y3: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( ( Y
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs3 @ ( cons @ A @ Y3 @ Ys4 ) ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_6157_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N3: nat,M3: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M3 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M3 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_6158_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A2: list @ A] :
( ( ( map @ A @ B @ F2 @ A2 )
= ( nil @ B ) )
= ( A2
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_6159_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F2 @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_6160_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_6161_list__update__nonempty,axiom,
! [A: $tType,Xs: list @ A,K: nat,X: A] :
( ( ( list_update @ A @ Xs @ K @ X )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% list_update_nonempty
thf(fact_6162_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_6163_Nil__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_in_shuffles
thf(fact_6164_num__of__nat__numeral__eq,axiom,
! [Q3: num] :
( ( num_of_nat @ ( numeral_numeral @ nat @ Q3 ) )
= Q3 ) ).
% num_of_nat_numeral_eq
thf(fact_6165_length__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( zero_zero @ nat ) )
= ( Xs
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_6166_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_6167_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_6168_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_6169_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( replicate @ A @ N @ X )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_6170_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_6171_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= ( nil @ nat ) )
= ( ( J
= ( zero_zero @ nat ) )
| ( ord_less_eq @ nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_6172_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F2: B > A,A2: A] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ ( nil @ B ) )
= ( zero_zero @ A ) ) ) ).
% horner_sum_simps(1)
thf(fact_6173_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_6174_length__greater__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( Xs
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
thf(fact_6175_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( cons @ nat @ ( numeral_numeral @ nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) )
& ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( nil @ nat ) ) ) ) ).
% upt_rec_numeral
thf(fact_6176_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F5: A > B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F5: A > B,A6: A,As: list @ A,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_6177_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P8: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( nil @ A ) ) )
=> ( ! [P8: A > A > $o,X3: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ~ ! [P8: A > A > $o,X3: A,Y3: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_6178_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F5: A > B,X3: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F5 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ( ! [F5: A > B,X3: A,Y3: A,Zs2: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F5 @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) ) )
=> ~ ! [A6: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A6 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_6179_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P8: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( nil @ A ) ) )
=> ~ ! [P8: A > A > $o,X3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X3 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_6180_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs3: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Xs3: list @ A,Y3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_6181_splice_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ~ ! [X3: A,Xs3: list @ A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_6182_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_6183_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X222: list @ A] :
( ( List
= ( cons @ A @ X21 @ X222 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_6184_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X223: list @ A] :
( Y
!= ( cons @ A @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_6185_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A] :
( ! [X3: A,Xs3: list @ A] :
( X
!= ( cons @ A @ X3 @ Xs3 ) )
=> ( X
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_6186_foldl__Nil,axiom,
! [A: $tType,B: $tType,F2: B > A > B,A2: B] :
( ( foldl @ B @ A @ F2 @ A2 @ ( nil @ A ) )
= A2 ) ).
% foldl_Nil
thf(fact_6187_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X3: A,Xs3: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_6188_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X3: A] :
( X
!= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Y3: A,Xs3: list @ A] :
( X
!= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_6189_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y4: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_6190_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs3: list @ A] : ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys4: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys4 ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_6191_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P @ Xs3 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_6192_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_6193_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_6194_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_6195_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B,Z: C,Zs2: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys4 ) @ ( cons @ C @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_6196_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D3: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,Ws: list @ D3,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D3 ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D3 ) @ Ws ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D3 ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: B,Ys4: list @ B,Z: C,Zs2: list @ C,W: D3,Ws2: list @ D3] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D3 ) @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ B @ Y3 @ Ys4 ) @ ( cons @ C @ Z @ Zs2 ) @ ( cons @ D3 @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_6197_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_6198_distinct__singleton,axiom,
! [A: $tType,X: A] : ( distinct @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_6199_sorted__wrt1,axiom,
! [A: $tType,P: A > A > $o,X: A] : ( sorted_wrt @ A @ P @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% sorted_wrt1
thf(fact_6200_list__update__code_I1_J,axiom,
! [A: $tType,I: nat,Y: A] :
( ( list_update @ A @ ( nil @ A ) @ I @ Y )
= ( nil @ A ) ) ).
% list_update_code(1)
thf(fact_6201_list__update_Osimps_I1_J,axiom,
! [A: $tType,I: nat,V2: A] :
( ( list_update @ A @ ( nil @ A ) @ I @ V2 )
= ( nil @ A ) ) ).
% list_update.simps(1)
thf(fact_6202_remove1_Osimps_I1_J,axiom,
! [A: $tType,X: A] :
( ( remove1 @ A @ X @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remove1.simps(1)
thf(fact_6203_product_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Uu3: list @ B] :
( ( product @ A @ B @ ( nil @ A ) @ Uu3 )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% product.simps(1)
thf(fact_6204_sorted__wrt_Osimps_I1_J,axiom,
! [A: $tType,P: A > A > $o] : ( sorted_wrt @ A @ P @ ( nil @ A ) ) ).
% sorted_wrt.simps(1)
thf(fact_6205_Nil__in__shufflesI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( nil @ A ) )
=> ( ( Ys
= ( nil @ A ) )
=> ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_6206_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_6207_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( map @ A @ B @ F2 @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_6208_shuffles_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( shuffles @ A @ Xs @ ( nil @ A ) )
= ( insert @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(2)
thf(fact_6209_shuffles_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( shuffles @ A @ ( nil @ A ) @ Ys )
= ( insert @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(1)
thf(fact_6210_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_6211_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_6212_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% strict_sorted_simps(1)
thf(fact_6213_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_6214_replicate__0,axiom,
! [A: $tType,X: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_6215_shufflesE,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys
!= ( nil @ A ) ) )
=> ( ( ( Zs = Ys )
=> ( Xs
!= ( nil @ A ) ) )
=> ( ! [X3: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs4 ) )
=> ! [Z: A,Zs4: list @ A] :
( ( Zs
= ( cons @ A @ Z @ Zs4 ) )
=> ( ( X3 = Z )
=> ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs4 @ Ys ) ) ) ) )
=> ~ ! [Y3: A,Ys5: list @ A] :
( ( Ys
= ( cons @ A @ Y3 @ Ys5 ) )
=> ! [Z: A,Zs4: list @ A] :
( ( Zs
= ( cons @ A @ Z @ Zs4 ) )
=> ( ( Y3 = Z )
=> ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_6216_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_6217_Bex__fold,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S8: $o] :
( S8
| ( P @ K3 ) )
@ $false
@ A3 ) ) ) ).
% Bex_fold
thf(fact_6218_find_Osimps_I1_J,axiom,
! [A: $tType,Uu3: A > $o] :
( ( find @ A @ Uu3 @ ( nil @ A ) )
= ( none @ A ) ) ).
% find.simps(1)
thf(fact_6219_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_6220_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_6221_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_6222_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_6223_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( bot_bot @ ( set @ B ) ) )
= ( nil @ B ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
thf(fact_6224_the__elem__set,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% the_elem_set
thf(fact_6225_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_6226_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_6227_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs: list @ A,F2: A > B,G: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_6228_upt__rec,axiom,
( upt
= ( ^ [I4: nat,J3: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I4 @ J3 ) @ ( cons @ nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_6229_shuffles_Opinduct,axiom,
! [A: $tType,A0: list @ A,A12: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A12 ) )
=> ( ! [Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( P @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) )
=> ( P @ Xs3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs3: list @ A,Y3: A,Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ( P @ Xs3 @ ( cons @ A @ Y3 @ Ys4 ) )
=> ( ( P @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) )
=> ( P @ A0 @ A12 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_6230_shuffles_Opsimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) )
=> ( ( shuffles @ A @ Xs @ ( nil @ A ) )
= ( insert @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(2)
thf(fact_6231_shuffles_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ( ( shuffles @ A @ ( nil @ A ) @ Ys )
= ( insert @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(1)
thf(fact_6232_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_6233_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_6234_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_6235_shuffles_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( insert @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( insert @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ~ ! [X3: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs3 ) )
=> ! [Y3: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( Y
!= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs3 @ ( cons @ A @ Y3 @ Ys4 ) ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_6236_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A3 )
= ( nil @ B ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
thf(fact_6237_Pow__set_I1_J,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( set2 @ A @ ( nil @ A ) ) )
= ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_set(1)
thf(fact_6238_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A4: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_max @ A @ X2 ) @ Y4 ) )
@ ( none @ A )
@ A4 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_6239_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_6240_set__Cons__sing__Nil,axiom,
! [A: $tType,A3: set @ A] :
( ( set_Cons @ A @ A3 @ ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
= ( image @ A @ ( list @ A )
@ ^ [X2: A] : ( cons @ A @ X2 @ ( nil @ A ) )
@ A3 ) ) ).
% set_Cons_sing_Nil
thf(fact_6241_transpose__rectangle,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I2 ) )
= N ) )
=> ( ( transpose @ A @ Xs )
= ( map @ nat @ ( list @ A )
@ ^ [I4: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I4 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_6242_transpose_Osimps_I2_J,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
= ( transpose @ A @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_6243_transpose_Osimps_I1_J,axiom,
! [A: $tType] :
( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ).
% transpose.simps(1)
thf(fact_6244_transpose__map__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ ( list @ B )] :
( ( transpose @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ Xs ) )
= ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ ( transpose @ B @ Xs ) ) ) ).
% transpose_map_map
thf(fact_6245_transpose__empty,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( ( transpose @ A @ Xs )
= ( nil @ ( list @ A ) ) )
= ( ! [X2: list @ A] :
( ( member @ ( list @ A ) @ X2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( X2
= ( nil @ A ) ) ) ) ) ).
% transpose_empty
thf(fact_6246_length__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs2: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ Xs
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_6247_listset_Osimps_I1_J,axiom,
! [A: $tType] :
( ( listset @ A @ ( nil @ ( set @ A ) ) )
= ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listset.simps(1)
thf(fact_6248_length__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( zero_zero @ nat ) ) )
& ( ( Xs
!= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% length_transpose_sorted
thf(fact_6249_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_6250_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_6251_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_6252_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_6253_set__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rev @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rev
thf(fact_6254_length__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rev
thf(fact_6255_distinct__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rev @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_rev
thf(fact_6256_rev__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( rev @ A @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) ).
% rev_replicate
thf(fact_6257_singleton__rev__conv,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( cons @ A @ X @ ( nil @ A ) )
= ( rev @ A @ Xs ) )
= ( ( cons @ A @ X @ ( nil @ A ) )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_6258_rev__singleton__conv,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ X @ ( nil @ A ) ) )
= ( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev_singleton_conv
thf(fact_6259_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_6260_rev__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( rev @ A @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( rev @ B @ Xs ) ) ) ).
% rev_map
thf(fact_6261_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_6262_sorted__wrt__rev,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( sorted_wrt @ A @ P @ ( rev @ A @ Xs ) )
= ( sorted_wrt @ A
@ ^ [X2: A,Y4: A] : ( P @ Y4 @ X2 )
@ Xs ) ) ).
% sorted_wrt_rev
thf(fact_6263_foldr__conv__foldl,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F3: B > A > A,Xs2: list @ B,A5: A] :
( foldl @ A @ B
@ ^ [X2: A,Y4: B] : ( F3 @ Y4 @ X2 )
@ A5
@ ( rev @ B @ Xs2 ) ) ) ) ).
% foldr_conv_foldl
thf(fact_6264_foldl__conv__foldr,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F3: A > B > A,A5: A,Xs2: list @ B] :
( foldr @ B @ A
@ ^ [X2: B,Y4: A] : ( F3 @ Y4 @ X2 )
@ ( rev @ B @ Xs2 )
@ A5 ) ) ) ).
% foldl_conv_foldr
thf(fact_6265_listset_Osimps_I2_J,axiom,
! [A: $tType,A3: set @ A,As2: list @ ( set @ A )] :
( ( listset @ A @ ( cons @ ( set @ A ) @ A3 @ As2 ) )
= ( set_Cons @ A @ A3 @ ( listset @ A @ As2 ) ) ) ).
% listset.simps(2)
thf(fact_6266_sorted__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) ) ) ).
% sorted_transpose
thf(fact_6267_rev__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_6268_rev__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y ) )
= ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y ) ) ) ).
% rev_update
thf(fact_6269_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I4 ) ) @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_6270_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_6271_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J3 ) @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_6272_foldr__max__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Y: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ( Xs
= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y )
= Y ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y )
= ( ord_max @ A @ ( nth @ A @ Xs @ ( zero_zero @ nat ) ) @ Y ) ) ) ) ) ) ).
% foldr_max_sorted
thf(fact_6273_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat,J: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( ord_less @ nat @ J
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I ) @ J )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J ) @ I ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_6274_transpose__column,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ).
% transpose_column
thf(fact_6275_filter__filter,axiom,
! [A: $tType,P: A > $o,Q: A > $o,Xs: list @ A] :
( ( filter2 @ A @ P @ ( filter2 @ A @ Q @ Xs ) )
= ( filter2 @ A
@ ^ [X2: A] :
( ( Q @ X2 )
& ( P @ X2 ) )
@ Xs ) ) ).
% filter_filter
thf(fact_6276_filter__True,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( ( filter2 @ A @ P @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_6277_remove1__filter__not,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ~ ( P @ X )
=> ( ( remove1 @ A @ X @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ).
% remove1_filter_not
thf(fact_6278_set__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( set2 @ A @ ( filter2 @ A @ P @ Xs ) )
= ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) ) ) ) ).
% set_filter
thf(fact_6279_filter__False,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X3 ) )
=> ( ( filter2 @ A @ P @ Xs )
= ( nil @ A ) ) ) ).
% filter_False
thf(fact_6280_filter_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( filter2 @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% filter.simps(1)
thf(fact_6281_filter__empty__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Xs )
= ( nil @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X2 ) ) ) ) ).
% filter_empty_conv
thf(fact_6282_empty__filter__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( nil @ A )
= ( filter2 @ A @ P @ Xs ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X2 ) ) ) ) ).
% empty_filter_conv
thf(fact_6283_filter__replicate,axiom,
! [A: $tType,P: A > $o,X: A,N: nat] :
( ( ( P @ X )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) )
& ( ~ ( P @ X )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) ) ) ).
% filter_replicate
thf(fact_6284_distinct__map__filter,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,P: B > $o] :
( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) )
=> ( distinct @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_6285_rev__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( rev @ A @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ ( rev @ A @ Xs ) ) ) ).
% rev_filter
thf(fact_6286_replicate__length__filter,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X )
@ Xs ) )
@ X )
= ( filter2 @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_6287_partition__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( member @ ( list @ A ) @ Xs
@ ( shuffles @ A @ ( filter2 @ A @ P @ Xs )
@ ( filter2 @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ Xs ) ) ) ).
% partition_in_shuffles
thf(fact_6288_sum__length__filter__compl,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) )
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ Xs ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_6289_inter__set__filter,axiom,
! [A: $tType,A3: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( set2 @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_6290_filter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Xs )
= Xs )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) ) ) ) ).
% filter_id_conv
thf(fact_6291_filter__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter2 @ A @ P @ Xs )
= ( filter2 @ A @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_6292_distinct__filter,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( filter2 @ A @ P @ Xs ) ) ) ).
% distinct_filter
thf(fact_6293_sorted__wrt__filter,axiom,
! [A: $tType,F2: A > A > $o,Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ F2 @ Xs )
=> ( sorted_wrt @ A @ F2 @ ( filter2 @ A @ P @ Xs ) ) ) ).
% sorted_wrt_filter
thf(fact_6294_filter__remove1,axiom,
! [A: $tType,Q: A > $o,X: A,Xs: list @ A] :
( ( filter2 @ A @ Q @ ( remove1 @ A @ X @ Xs ) )
= ( remove1 @ A @ X @ ( filter2 @ A @ Q @ Xs ) ) ) ).
% filter_remove1
thf(fact_6295_length__filter__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_filter_le
thf(fact_6296_filter__is__subset,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( filter2 @ A @ P @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% filter_is_subset
thf(fact_6297_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G: ( list @ A ) > A,Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X2: A] :
( X2
= ( G @ Xs ) )
@ Xs ) ) ) ).
% sorted_same
thf(fact_6298_filter_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( filter2 @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( filter2 @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( filter2 @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_6299_filter__shuffles,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] :
( ( image @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( shuffles @ A @ ( filter2 @ A @ P @ Xs ) @ ( filter2 @ A @ P @ Ys ) ) ) ).
% filter_shuffles
thf(fact_6300_length__filter__less,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_6301_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,P: B > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ).
% sorted_filter
thf(fact_6302_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,G: ( list @ B ) > A,Xs: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F2
@ ( filter2 @ B
@ ^ [X2: B] :
( ( F2 @ X2 )
= ( G @ Xs ) )
@ Xs ) ) ) ) ).
% sorted_map_same
thf(fact_6303_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F2: B > A,P: B > $o,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_6304_sum__list__filter__le__nat,axiom,
! [A: $tType,F2: A > nat,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ ( filter2 @ A @ P @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) ) ).
% sum_list_filter_le_nat
thf(fact_6305_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B,P: B > $o,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ( F2 @ X3 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_6306_set__minus__filter__out,axiom,
! [A: $tType,Xs: list @ A,Y: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X2: A] : X2 != Y
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_6307_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_6308_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_6309_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_6310_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_6311_length__filter__conv__card,axiom,
! [A: $tType,P5: A > $o,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P5 @ Xs ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P5 @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_6312_distinct__length__filter,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) )
= ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_length_filter
thf(fact_6313_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Xss2: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs2: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) )
@ Xss2
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X2: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss2 )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_6314_nth__transpose,axiom,
! [A: $tType,I: nat,Xs: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs2: list @ A] : ( nth @ A @ Xs2 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_6315_transpose__max__length,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs2: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ ( transpose @ A @ Xs )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [X2: list @ A] :
( X2
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_6316_transpose__column__length,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ) ).
% transpose_column_length
thf(fact_6317_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F2: B > A,P: B > $o,Xs: list @ B] :
( ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) )
= ( map_filter @ B @ A
@ ^ [X2: B] : ( if @ ( option @ A ) @ ( P @ X2 ) @ ( some @ A @ ( F2 @ X2 ) ) @ ( none @ A ) )
@ Xs ) ) ).
% map_filter_map_filter
thf(fact_6318_transpose__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( transpose @ A @ ( transpose @ A @ Xs ) )
= ( takeWhile @ ( list @ A )
@ ^ [X2: list @ A] :
( X2
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_transpose
thf(fact_6319_takeWhile__idem,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( takeWhile @ A @ P @ ( takeWhile @ A @ P @ Xs ) )
= ( takeWhile @ A @ P @ Xs ) ) ).
% takeWhile_idem
thf(fact_6320_takeWhile__eq__all__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( takeWhile @ A @ P @ Xs )
= Xs )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) ) ) ) ).
% takeWhile_eq_all_conv
thf(fact_6321_takeWhile__replicate,axiom,
! [A: $tType,P: A > $o,X: A,N: nat] :
( ( ( P @ X )
=> ( ( takeWhile @ A @ P @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) )
& ( ~ ( P @ X )
=> ( ( takeWhile @ A @ P @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) ) ) ).
% takeWhile_replicate
thf(fact_6322_takeWhile_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( takeWhile @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( takeWhile @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( takeWhile @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) ) ) ).
% takeWhile.simps(2)
thf(fact_6323_sorted__takeWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% sorted_takeWhile
thf(fact_6324_length__takeWhile__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_takeWhile_le
thf(fact_6325_distinct__takeWhile,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( takeWhile @ A @ P @ Xs ) ) ) ).
% distinct_takeWhile
thf(fact_6326_set__takeWhileD,axiom,
! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( takeWhile @ A @ P @ Xs ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( P @ X ) ) ) ).
% set_takeWhileD
thf(fact_6327_takeWhile__cong,axiom,
! [A: $tType,L: list @ A,K: list @ A,P: A > $o,Q: A > $o] :
( ( L = K )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ L ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( takeWhile @ A @ P @ L )
= ( takeWhile @ A @ Q @ K ) ) ) ) ).
% takeWhile_cong
thf(fact_6328_takeWhile_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( takeWhile @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% takeWhile.simps(1)
thf(fact_6329_map__filter__simps_I2_J,axiom,
! [B: $tType,A: $tType,F2: B > ( option @ A )] :
( ( map_filter @ B @ A @ F2 @ ( nil @ B ) )
= ( nil @ A ) ) ).
% map_filter_simps(2)
thf(fact_6330_takeWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) )
=> ( ( nth @ A @ ( takeWhile @ A @ P @ Xs ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% takeWhile_nth
thf(fact_6331_nth__length__takeWhile,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_6332_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),X: B,Xs: list @ B] :
( ( map_filter @ B @ A @ F2 @ ( cons @ B @ X @ Xs ) )
= ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F2 @ Xs )
@ ^ [Y4: A] : ( cons @ A @ Y4 @ ( map_filter @ B @ A @ F2 @ Xs ) )
@ ( F2 @ X ) ) ) ).
% map_filter_simps(1)
thf(fact_6333_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( P @ ( nth @ A @ Xs @ I2 ) ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_6334_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F2 @ Xs ) ) )
=> ( ( filter2 @ B
@ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F2 @ X2 ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F2 @ X2 ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_6335_Set__filter__fold,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( filter3 @ A @ P @ A3 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X2: A,A14: set @ A] : ( if @ ( set @ A ) @ ( P @ X2 ) @ ( insert @ A @ X2 @ A14 ) @ A14 )
@ ( bot_bot @ ( set @ A ) )
@ A3 ) ) ) ).
% Set_filter_fold
thf(fact_6336_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_6337_member__filter,axiom,
! [A: $tType,X: A,P: A > $o,A3: set @ A] :
( ( member @ A @ X @ ( filter3 @ A @ P @ A3 ) )
= ( ( member @ A @ X @ A3 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_6338_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P3: A > $o,A4: set @ A] :
( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A4 )
& ( P3 @ A5 ) ) ) ) ) ).
% Set.filter_def
thf(fact_6339_filter__set,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( filter3 @ A @ P @ ( set2 @ A @ Xs ) )
= ( set2 @ A @ ( filter2 @ A @ P @ Xs ) ) ) ).
% filter_set
thf(fact_6340_inter__Set__filter,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( filter3 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 )
@ B3 ) ) ) ).
% inter_Set_filter
thf(fact_6341_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) )
=> ( ( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) )
& ( ~ ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ) ) ).
% upto.psimps
thf(fact_6342_upto_Opelims,axiom,
! [X: int,Xa: int,Y: list @ int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_6343_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= ( nil @ int ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil
thf(fact_6344_upto__Nil2,axiom,
! [I: int,J: int] :
( ( ( nil @ int )
= ( upto @ I @ J ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil2
thf(fact_6345_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less @ int @ J @ I )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_6346_upto__single,axiom,
! [I: int] :
( ( upto @ I @ I )
= ( cons @ int @ I @ ( nil @ int ) ) ) ).
% upto_single
thf(fact_6347_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J )
=> ( ( nth @ int @ ( upto @ I @ J ) @ K )
= ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% nth_upto
thf(fact_6348_length__upto,axiom,
! [I: int,J: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_6349_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(1)
thf(fact_6350_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(4)
thf(fact_6351_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(3)
thf(fact_6352_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(2)
thf(fact_6353_upto__code,axiom,
( upto
= ( ^ [I4: int,J3: int] : ( upto_aux @ I4 @ J3 @ ( nil @ int ) ) ) ) ).
% upto_code
thf(fact_6354_sorted__upto,axiom,
! [M: int,N: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M @ N ) ) ).
% sorted_upto
thf(fact_6355_sorted__wrt__upto,axiom,
! [I: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I @ J ) ) ).
% sorted_wrt_upto
thf(fact_6356_distinct__upto,axiom,
! [I: int,J: int] : ( distinct @ int @ ( upto @ I @ J ) ) ).
% distinct_upto
thf(fact_6357_atLeastAtMost__upto,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I4: int,J3: int] : ( set2 @ int @ ( upto @ I4 @ J3 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_6358_atLeastLessThan__upto,axiom,
( ( set_or7035219750837199246ssThan @ int )
= ( ^ [I4: int,J3: int] : ( set2 @ int @ ( upto @ I4 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_6359_upto_Osimps,axiom,
( upto
= ( ^ [I4: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I4 @ J3 ) @ ( cons @ int @ I4 @ ( upto @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_6360_upto_Oelims,axiom,
! [X: int,Xa: int,Y: list @ int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_6361_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_6362_splice_Opinduct,axiom,
! [A: $tType,A0: list @ A,A12: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A12 ) )
=> ( ! [Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( P @ ( nil @ A ) @ Ys4 ) )
=> ( ! [X3: A,Xs3: list @ A,Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 ) )
=> ( ( P @ Ys4 @ Xs3 )
=> ( P @ ( cons @ A @ X3 @ Xs3 ) @ Ys4 ) ) )
=> ( P @ A0 @ A12 ) ) ) ) ).
% splice.pinduct
thf(fact_6363_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A4: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( sup_sup @ A @ X2 ) @ Y4 ) )
@ ( none @ A )
@ A4 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_6364_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Sup_fin.singleton
thf(fact_6365_inf__Sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= A2 ) ) ) ) ).
% inf_Sup_absorb
thf(fact_6366_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A3 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_6367_Sup__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_6368_Sup__fin__Max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% Sup_fin_Max
thf(fact_6369_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_6370_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_6371_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ A @ A6 @ X ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_6372_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X )
=> ! [A11: A] :
( ( member @ A @ A11 @ A3 )
=> ( ord_less_eq @ A @ A11 @ X ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_6373_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Sup_fin.infinite
thf(fact_6374_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B3 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_6375_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H2: A > A,N4: set @ A] :
( ! [X3: A,Y3: A] :
( ( H2 @ ( sup_sup @ A @ X3 @ Y3 ) )
= ( sup_sup @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic5882676163264333800up_fin @ A @ N4 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image @ A @ A @ H2 @ N4 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_6376_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B3 ) @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_6377_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_6378_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A3 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_6379_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B3 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_6380_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A3 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X @ A3 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_6381_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A3 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_6382_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_6383_splice_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [X3: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs3 ) )
=> ( ( Y
= ( cons @ A @ X3 @ ( splice @ A @ Xa @ Xs3 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs3 ) @ Xa ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_6384_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A4: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( inf_inf @ A @ X2 ) @ Y4 ) )
@ ( none @ A )
@ A4 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_6385_split__Nil__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( splice @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% split_Nil_iff
thf(fact_6386_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_6387_splice__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] : ( member @ ( list @ A ) @ ( splice @ A @ Xs @ Ys ) @ ( shuffles @ A @ Xs @ Ys ) ) ).
% splice_in_shuffles
thf(fact_6388_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Inf_fin.singleton
thf(fact_6389_sup__Inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ A2 )
= A2 ) ) ) ) ).
% sup_Inf_absorb
thf(fact_6390_length__splice,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( size_size @ ( list @ A ) @ ( splice @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% length_splice
thf(fact_6391_splice__replicate,axiom,
! [A: $tType,M: nat,X: A,N: nat] :
( ( splice @ A @ ( replicate @ A @ M @ X ) @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ ( plus_plus @ nat @ M @ N ) @ X ) ) ).
% splice_replicate
thf(fact_6392_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A3 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_6393_splice_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( splice @ A @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_6394_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_6395_Inf__fin__Min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% Inf_fin_Min
thf(fact_6396_Inf__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ A2 ) ) ) ) ).
% Inf_fin.coboundedI
thf(fact_6397_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_6398_splice_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [X3: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs3 ) )
=> ( Y
!= ( cons @ A @ X3 @ ( splice @ A @ Xa @ Xs3 ) ) ) ) ) ) ).
% splice.elims
thf(fact_6399_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
=> ! [A11: A] :
( ( member @ A @ A11 @ A3 )
=> ( ord_less_eq @ A @ X @ A11 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_6400_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ A @ X @ A6 ) )
=> ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_6401_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_6402_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Inf_fin.infinite
thf(fact_6403_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B3 ) @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_6404_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H2: A > A,N4: set @ A] :
( ! [X3: A,Y3: A] :
( ( H2 @ ( inf_inf @ A @ X3 @ Y3 ) )
= ( inf_inf @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic7752659483105999362nf_fin @ A @ N4 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image @ A @ A @ H2 @ N4 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_6405_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B3 ) @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_6406_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A3 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_6407_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_6408_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B3 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic7752659483105999362nf_fin @ A @ B3 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_6409_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_6410_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A3 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X @ A3 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_6411_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_6412_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A3 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A3 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_6413_splice_Opsimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) )
=> ( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( splice @ A @ Ys @ Xs ) ) ) ) ).
% splice.psimps(2)
thf(fact_6414_splice_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ) ).
% splice.psimps(1)
thf(fact_6415_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q6: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_6416_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( linord4507533701916653071of_set @ A @ A3 )
= ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_6417_sorted__list__of__set__range,axiom,
! [M: nat,N: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sorted_list_of_set_range
thf(fact_6418_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( nil @ A ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_6419_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ A3 )
= ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_6420_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A3 ) )
= A3 ) ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_6421_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( size_size @ ( list @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) )
= ( finite_card @ A @ A3 ) ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_6422_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( linord4507533701916653071of_set @ A @ A3 )
= ( nil @ A ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_6423_sgn__integer__code,axiom,
( ( sgn_sgn @ code_integer )
= ( ^ [K3: code_integer] :
( if @ code_integer
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% sgn_integer_code
thf(fact_6424_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( ( linord4507533701916653071of_set @ A @ A3 )
= ( linord4507533701916653071of_set @ A @ B3 ) )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( A3 = B3 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_6425_less__eq__integer__code_I1_J,axiom,
ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ).
% less_eq_integer_code(1)
thf(fact_6426_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M3: num,N3: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M3 ) @ ( numeral_numeral @ code_integer @ N3 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M3 ) @ ( numeral_numeral @ code_integer @ N3 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_6427_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] : ( distinct @ A @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_6428_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_6429_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_6430_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_6431_zero__integer_Orsp,axiom,
( ( zero_zero @ int )
= ( zero_zero @ int ) ) ).
% zero_integer.rsp
thf(fact_6432_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs ) )
= Xs ) ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_6433_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_6434_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,L: list @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A3 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A3 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A3 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_6435_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_6436_Code__Numeral_Opositive__def,axiom,
( code_positive
= ( numeral_numeral @ code_integer ) ) ).
% Code_Numeral.positive_def
thf(fact_6437_abs__integer__code,axiom,
( ( abs_abs @ code_integer )
= ( ^ [K3: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K3 ) @ K3 ) ) ) ).
% abs_integer_code
thf(fact_6438_less__integer__code_I1_J,axiom,
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(1)
thf(fact_6439_zero__integer__def,axiom,
( ( zero_zero @ code_integer )
= ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% zero_integer_def
thf(fact_6440_less__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( ord_less @ int @ Xa @ X ) ) ).
% less_integer.abs_eq
thf(fact_6441_less__eq__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( ord_less_eq @ int @ Xa @ X ) ) ).
% less_eq_integer.abs_eq
thf(fact_6442_integer__of__num_I3_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit1 @ N ) )
= ( plus_plus @ code_integer @ ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% integer_of_num(3)
thf(fact_6443_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K3: code_integer] :
( if @ int @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K3 ) ) )
@ ( if @ int
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ int )
@ ( product_case_prod @ code_integer @ code_integer @ int
@ ^ [L2: code_integer,J3: code_integer] :
( if @ int
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) )
@ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) ) @ ( one_one @ int ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_6444_int__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ).
% int_of_integer_of_nat
thf(fact_6445_zero__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ int ) ) ).
% zero_integer.rep_eq
thf(fact_6446_int__of__integer__numeral,axiom,
! [K: num] :
( ( code_int_of_integer @ ( numeral_numeral @ code_integer @ K ) )
= ( numeral_numeral @ int @ K ) ) ).
% int_of_integer_numeral
thf(fact_6447_less__integer_Orep__eq,axiom,
( ( ord_less @ code_integer )
= ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_6448_integer__less__iff,axiom,
( ( ord_less @ code_integer )
= ( ^ [K3: code_integer,L2: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_less_iff
thf(fact_6449_less__eq__integer_Orep__eq,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_eq_integer.rep_eq
thf(fact_6450_integer__less__eq__iff,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_less_eq_iff
thf(fact_6451_integer__of__num__def,axiom,
( code_integer_of_num
= ( numeral_numeral @ code_integer ) ) ).
% integer_of_num_def
thf(fact_6452_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one2 )
= ( one_one @ code_integer ) ) ).
% integer_of_num_triv(1)
thf(fact_6453_integer__of__num_I2_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit0 @ N ) )
= ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% integer_of_num(2)
thf(fact_6454_integer__of__num__triv_I2_J,axiom,
( ( code_integer_of_num @ ( bit0 @ one2 ) )
= ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% integer_of_num_triv(2)
thf(fact_6455_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K3: code_integer] :
( if @ num @ ( ord_less_eq @ code_integer @ K3 @ ( one_one @ code_integer ) ) @ one2
@ ( product_case_prod @ code_integer @ code_integer @ num
@ ^ [L2: code_integer,J3: code_integer] :
( if @ num
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) )
@ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one2 ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_6456_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K3: code_integer] :
( if @ nat @ ( ord_less_eq @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
@ ( product_case_prod @ code_integer @ code_integer @ nat
@ ^ [L2: code_integer,J3: code_integer] :
( if @ nat
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( one_one @ nat ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_6457_nat__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
= N ) ).
% nat_of_integer_of_nat
thf(fact_6458_of__nat__of__integer,axiom,
! [K: code_integer] :
( ( semiring_1_of_nat @ code_integer @ ( code_nat_of_integer @ K ) )
= ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% of_nat_of_integer
thf(fact_6459_nat__of__integer__non__positive,axiom,
! [K: code_integer] :
( ( ord_less_eq @ code_integer @ K @ ( zero_zero @ code_integer ) )
=> ( ( code_nat_of_integer @ K )
= ( zero_zero @ nat ) ) ) ).
% nat_of_integer_non_positive
thf(fact_6460_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_6461_nat__of__integer__code__post_I3_J,axiom,
! [K: num] :
( ( code_nat_of_integer @ ( numeral_numeral @ code_integer @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_of_integer_code_post(3)
thf(fact_6462_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
@ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K3 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_6463_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ~ ( ( ( Nat
= ( zero_zero @ nat ) )
& ~ ( P @ F1 ) )
| ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
& ~ ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_6464_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X24: nat] : X24 ) ) ).
% pred_def
thf(fact_6465_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ( ( Nat
= ( zero_zero @ nat ) )
=> ( P @ F1 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_6466_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( if @ ( product_prod @ code_integer @ $o )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
@ ^ [R5: code_integer,S8: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ R5 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ S8 ) )
@ ( S8
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_6467_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X2: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X2 ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_6468_linorder_OMax_Ocong,axiom,
! [A: $tType] :
( ( lattices_Max @ A )
= ( lattices_Max @ A ) ) ).
% linorder.Max.cong
thf(fact_6469_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L2 )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S8: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S8
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L2 @ S8 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S8: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S8
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L2 ) @ S8 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_6470_finite__enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S2 ) )
=> ( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S8: A] :
( ( member @ A @ S8 @ S2 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ S8 ) ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_6471_enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,M: nat,N: nat] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M ) @ ( infini527867602293511546merate @ A @ S2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ).
% enumerate_mono_iff
thf(fact_6472_finite__enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,M: nat,N: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ M @ ( finite_card @ A @ S2 ) )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S2 ) )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M ) @ ( infini527867602293511546merate @ A @ S2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_6473_le__enumerate,axiom,
! [S2: set @ nat,N: nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S2 @ N ) ) ) ).
% le_enumerate
thf(fact_6474_enumerate__0,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A] :
( ( infini527867602293511546merate @ A @ S2 @ ( zero_zero @ nat ) )
= ( ord_Least @ A
@ ^ [N3: A] : ( member @ A @ N3 @ S2 ) ) ) ) ).
% enumerate_0
thf(fact_6475_enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) ) ) ) ) ).
% enumerate_step
thf(fact_6476_enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M: nat,N: nat,S2: set @ A] :
( ( ord_less @ nat @ M @ N )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M ) @ ( infini527867602293511546merate @ A @ S2 @ N ) ) ) ) ) ).
% enumerate_mono
thf(fact_6477_finite__enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S2 ) )
=> ( member @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ S2 ) ) ) ) ).
% finite_enumerate_in_set
thf(fact_6478_finite__enumerate__Ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,S: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( member @ A @ S @ S2 )
=> ? [N2: nat] :
( ( ord_less @ nat @ N2 @ ( finite_card @ A @ S2 ) )
& ( ( infini527867602293511546merate @ A @ S2 @ N2 )
= S ) ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_6479_finite__enum__ext,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X5: set @ A,Y6: set @ A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( finite_card @ A @ X5 ) )
=> ( ( infini527867602293511546merate @ A @ X5 @ I2 )
= ( infini527867602293511546merate @ A @ Y6 @ I2 ) ) )
=> ( ( finite_finite2 @ A @ X5 )
=> ( ( finite_finite2 @ A @ Y6 )
=> ( ( ( finite_card @ A @ X5 )
= ( finite_card @ A @ Y6 ) )
=> ( X5 = Y6 ) ) ) ) ) ) ).
% finite_enum_ext
thf(fact_6480_finite__enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M: nat,N: nat,S2: set @ A] :
( ( ord_less @ nat @ M @ N )
=> ( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S2 ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M ) @ ( infini527867602293511546merate @ A @ S2 @ N ) ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_6481_finite__le__enumerate,axiom,
! [S2: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ S2 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ nat @ S2 ) )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S2 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_6482_enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S8: A] :
( ( member @ A @ S8 @ S2 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ S8 ) ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_6483_finite__enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S2 ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_6484_enumerate__Suc_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ ( infini527867602293511546merate @ A @ S2 @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N ) ) ) ).
% enumerate_Suc'
thf(fact_6485_finite__enum__subset,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X5: set @ A,Y6: set @ A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( finite_card @ A @ X5 ) )
=> ( ( infini527867602293511546merate @ A @ X5 @ I2 )
= ( infini527867602293511546merate @ A @ Y6 @ I2 ) ) )
=> ( ( finite_finite2 @ A @ X5 )
=> ( ( finite_finite2 @ A @ Y6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ X5 ) @ ( finite_card @ A @ Y6 ) )
=> ( ord_less_eq @ ( set @ A ) @ X5 @ Y6 ) ) ) ) ) ) ).
% finite_enum_subset
thf(fact_6486_finite__enumerate__initial__segment,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat,S: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ S2 @ ( set_ord_lessThan @ A @ S ) ) ) )
=> ( ( infini527867602293511546merate @ A @ ( inf_inf @ ( set @ A ) @ S2 @ ( set_ord_lessThan @ A @ S ) ) @ N )
= ( infini527867602293511546merate @ A @ S2 @ N ) ) ) ) ) ).
% finite_enumerate_initial_segment
thf(fact_6487_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S2
@ ( insert @ A
@ ( ord_Least @ A
@ ^ [N3: A] : ( member @ A @ N3 @ S2 ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N ) ) ) ).
% enumerate_Suc
thf(fact_6488_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X2: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X2 ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_6489_rec__nat__add__eq__if,axiom,
! [A: $tType,A2: A,F2: nat > A > A,V2: num,N: nat] :
( ( rec_nat @ A @ A2 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) @ ( rec_nat @ A @ A2 @ F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) ) ) ) ).
% rec_nat_add_eq_if
thf(fact_6490_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_6491_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_6492_rec__nat__numeral,axiom,
! [A: $tType,A2: A,F2: nat > A > A,V2: num] :
( ( rec_nat @ A @ A2 @ F2 @ ( numeral_numeral @ nat @ V2 ) )
= ( F2 @ ( pred_numeral @ V2 ) @ ( rec_nat @ A @ A2 @ F2 @ ( pred_numeral @ V2 ) ) ) ) ).
% rec_nat_numeral
thf(fact_6493_linorder_OMin_Ocong,axiom,
! [A: $tType] :
( ( lattices_Min @ A )
= ( lattices_Min @ A ) ) ).
% linorder.Min.cong
thf(fact_6494_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F12: T,F23: nat > T > T,X2: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F23 @ X2 ) ) ) ) ).
% old.rec_nat_def
thf(fact_6495_rec__nat__0__imp,axiom,
! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A] :
( ( F2
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F2 @ ( zero_zero @ nat ) )
= F1 ) ) ).
% rec_nat_0_imp
thf(fact_6496_subset__CollectI,axiom,
! [A: $tType,B3: set @ A,A3: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B3 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ B3 )
& ( Q @ X2 ) ) )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_6497_subset__Collect__iff,axiom,
! [A: $tType,B3: set @ A,A3: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B3 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_6498_rec__nat__Suc__imp,axiom,
! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A,N: nat] :
( ( F2
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F2 @ ( suc @ N ) )
= ( F22 @ N @ ( F2 @ N ) ) ) ) ).
% rec_nat_Suc_imp
thf(fact_6499_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_6500_drop__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_6501_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_6502_drop__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A2 ) ) ) ).
% drop_bit_drop_bit
thf(fact_6503_drop__bit__and,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) @ ( bit_se4197421643247451524op_bit @ A @ N @ B2 ) ) ) ) ).
% drop_bit_and
thf(fact_6504_drop__bit__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) @ ( bit_se4197421643247451524op_bit @ A @ N @ B2 ) ) ) ) ).
% drop_bit_or
thf(fact_6505_drop__bit__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) @ ( bit_se4197421643247451524op_bit @ A @ N @ B2 ) ) ) ) ).
% drop_bit_xor
thf(fact_6506_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_6507_drop__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% drop_bit_nonnegative_int_iff
thf(fact_6508_drop__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% drop_bit_negative_int_iff
thf(fact_6509_drop__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% drop_bit_minus_one
thf(fact_6510_drop__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit0
thf(fact_6511_drop__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit1
thf(fact_6512_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_6513_drop__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_numeral_bit0
thf(fact_6514_drop__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_numeral_bit1
thf(fact_6515_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_6516_drop__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_6517_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_6518_drop__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N @ M ) @ ( bit_se4197421643247451524op_bit @ A @ M @ A2 ) ) ) ) ).
% drop_bit_take_bit
thf(fact_6519_drop__bit__push__bit__int,axiom,
! [M: nat,N: nat,K: int] :
( ( bit_se4197421643247451524op_bit @ int @ M @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) )
= ( bit_se4197421643247451524op_bit @ int @ ( minus_minus @ nat @ M @ N ) @ ( bit_se4730199178511100633sh_bit @ int @ ( minus_minus @ nat @ N @ M ) @ K ) ) ) ).
% drop_bit_push_bit_int
thf(fact_6520_take__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A2 ) ) ) ) ).
% take_bit_drop_bit
thf(fact_6521_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= A2 )
= ( ( bit_se4197421643247451524op_bit @ A @ N @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_6522_of__nat__drop__bit,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ M @ N ) )
= ( bit_se4197421643247451524op_bit @ A @ M @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_drop_bit
thf(fact_6523_drop__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,M: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ N @ M ) ) ) ) ).
% drop_bit_of_nat
thf(fact_6524_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( divide_divide @ A @ A2 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_6525_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A5 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_6526_bits__ident,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= A2 ) ) ).
% bits_ident
thf(fact_6527_drop__bit__half,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% drop_bit_half
thf(fact_6528_stable__imp__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( bit_se4197421643247451524op_bit @ A @ N @ A2 )
= A2 ) ) ) ).
% stable_imp_drop_bit_eq
thf(fact_6529_drop__bit__int__def,axiom,
( ( bit_se4197421643247451524op_bit @ int )
= ( ^ [N3: nat,K3: int] : ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% drop_bit_int_def
thf(fact_6530_drop__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ A2 )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% drop_bit_Suc
thf(fact_6531_drop__bit__eq__div,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N3: nat,A5: A] : ( divide_divide @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% drop_bit_eq_div
thf(fact_6532_bit__iff__odd__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N3: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A5 ) ) ) ) ) ).
% bit_iff_odd_drop_bit
thf(fact_6533_even__drop__bit__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% even_drop_bit_iff_not_bit
thf(fact_6534_slice__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ).
% slice_eq_mask
thf(fact_6535_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N3: nat,A5: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ A5
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_6536_max__extp_Omax__extI,axiom,
! [A: $tType,X5: set @ A,Y6: set @ A,R2: A > A > $o] :
( ( finite_finite2 @ A @ X5 )
=> ( ( finite_finite2 @ A @ Y6 )
=> ( ( Y6
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Y6 )
& ( R2 @ X3 @ Xa2 ) ) )
=> ( max_extp @ A @ R2 @ X5 @ Y6 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_6537_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_6538_drop__bit__nat__eq,axiom,
! [N: nat,K: int] :
( ( bit_se4197421643247451524op_bit @ nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) ) ) ).
% drop_bit_nat_eq
thf(fact_6539_drop__bit__nat__def,axiom,
( ( bit_se4197421643247451524op_bit @ nat )
= ( ^ [N3: nat,M3: nat] : ( divide_divide @ nat @ M3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% drop_bit_nat_def
thf(fact_6540_max__extp_Ocases,axiom,
! [A: $tType,R2: A > A > $o,A12: set @ A,A23: set @ A] :
( ( max_extp @ A @ R2 @ A12 @ A23 )
=> ~ ( ( finite_finite2 @ A @ A12 )
=> ( ( finite_finite2 @ A @ A23 )
=> ( ( A23
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X6: A] :
( ( member @ A @ X6 @ A12 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A23 )
& ( R2 @ X6 @ Xa3 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_6541_max__extp_Osimps,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R6: A > A > $o,A13: set @ A,A24: set @ A] :
( ( finite_finite2 @ A @ A13 )
& ( finite_finite2 @ A @ A24 )
& ( A24
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A13 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A24 )
& ( R6 @ X2 @ Y4 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_6542_Suc__0__mod__numeral,axiom,
! [K: num] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
= ( product_snd @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% Suc_0_mod_numeral
thf(fact_6543_Suc__0__div__numeral,axiom,
! [K: num] :
( ( divide_divide @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
= ( product_fst @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% Suc_0_div_numeral
thf(fact_6544_numeral__div__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [K: num,L: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ A @ L ) )
= ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ K @ L ) ) ) ) ).
% numeral_div_numeral
thf(fact_6545_numeral__mod__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [K: num,L: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ A @ L ) )
= ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ K @ L ) ) ) ) ).
% numeral_mod_numeral
thf(fact_6546_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_6547_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_6548_pair__list__eqI,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs )
= ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) )
=> ( ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs )
= ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_6549_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X5: set @ A,A3: set @ ( product_prod @ A @ B ),Y6: set @ B,P: A > B > $o,Q: A > B > $o] :
( ( X5
= ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) )
=> ( ( Y6
= ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ Y6 )
=> ( ( P @ X3 @ Xa3 )
=> ( Q @ X3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_6550_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C6: A > B > C,P6: product_prod @ A @ B] : ( C6 @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) ) ) ).
% case_prod_unfold
thf(fact_6551_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F3: A > B > C,X2: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ X2 ) @ ( product_snd @ A @ B @ X2 ) ) ) ) ).
% case_prod_beta'
thf(fact_6552_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D3: $tType,F2: A > B > C,G: D3 > A] :
( ( ^ [U: product_prod @ D3 @ B] : ( F2 @ ( G @ ( product_fst @ D3 @ B @ U ) ) @ ( product_snd @ D3 @ B @ U ) ) )
= ( product_case_prod @ D3 @ B @ C
@ ^ [X2: D3] : ( F2 @ ( G @ X2 ) ) ) ) ).
% split_comp_eq
thf(fact_6553_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,P5: product_prod @ B @ A,Q: C > $o,F2: B > A > C] :
( ( P @ ( product_snd @ B @ A @ P5 ) @ ( product_fst @ B @ A @ P5 ) )
=> ( ! [X3: B,Y3: A] :
( ( P @ Y3 @ X3 )
=> ( Q @ ( F2 @ X3 @ Y3 ) ) )
=> ( Q @ ( product_case_prod @ B @ A @ C @ F2 @ P5 ) ) ) ) ).
% exE_realizer
thf(fact_6554_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X12: A,X24: B] : X24 ) ) ).
% snd_def
thf(fact_6555_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X12: A,X24: B] : X12 ) ) ).
% fst_def
thf(fact_6556_The__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_6557_Eps__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( fChoice @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
= ( fChoice @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% Eps_case_prod
thf(fact_6558_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_6559_fst__divmod,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ M @ N ) )
= ( divide_divide @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% fst_divmod
thf(fact_6560_snd__divmod,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ M @ N ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% snd_divmod
thf(fact_6561_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F3: A > nat,G2: B > nat,P6: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F3 @ ( product_fst @ A @ B @ P6 ) ) @ ( G2 @ ( product_snd @ A @ B @ P6 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_6562_in__set__enumerate__eq,axiom,
! [A: $tType,P5: product_prod @ nat @ A,N: nat,Xs: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P5 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) )
= ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P5 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P5 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
& ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P5 ) @ N ) )
= ( product_snd @ nat @ A @ P5 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_6563_enumerate__simps_I1_J,axiom,
! [A: $tType,N: nat] :
( ( enumerate @ A @ N @ ( nil @ A ) )
= ( nil @ ( product_prod @ nat @ A ) ) ) ).
% enumerate_simps(1)
thf(fact_6564_length__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( product_prod @ nat @ A ) ) @ ( enumerate @ A @ N @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_enumerate
thf(fact_6565_map__snd__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ A @ ( product_snd @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) )
= Xs ) ).
% map_snd_enumerate
thf(fact_6566_enumerate__simps_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list @ B] :
( ( enumerate @ B @ N @ ( cons @ B @ X @ Xs ) )
= ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X ) @ ( enumerate @ B @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_6567_map__fst__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) )
= ( upt @ N @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% map_fst_enumerate
thf(fact_6568_distinct__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( distinct @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ).
% distinct_enumerate
thf(fact_6569_quotient__of__denom__pos_H,axiom,
! [R3: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ ( quotient_of @ R3 ) ) ) ).
% quotient_of_denom_pos'
thf(fact_6570_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_6571_bezw_Osimps,axiom,
( bezw
= ( ^ [X2: 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 @ X2 @ Y4 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y4 @ ( modulo_modulo @ nat @ X2 @ Y4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y4 @ ( modulo_modulo @ nat @ X2 @ Y4 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Y4 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_6572_bezw_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_6573_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F2: nat > A,M: nat] :
( ( enumerate @ A @ N @ ( map @ nat @ A @ F2 @ ( upt @ N @ M ) ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [K3: nat] : ( product_Pair @ nat @ A @ K3 @ ( F2 @ K3 ) )
@ ( upt @ N @ M ) ) ) ).
% enumerate_map_upt
thf(fact_6574_sorted__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% sorted_enumerate
thf(fact_6575_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M: nat,A2: A] :
( ( enumerate @ A @ N @ ( replicate @ A @ M @ A2 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q6: nat] : ( product_Pair @ nat @ A @ Q6 @ A2 )
@ ( upt @ N @ ( plus_plus @ nat @ N @ M ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_6576_nth__enumerate__eq,axiom,
! [A: $tType,M: nat,Xs: list @ A,N: nat] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M ) @ ( nth @ A @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_6577_bezw_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% bezw.pelims
thf(fact_6578_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_6579_minus__numeral__mod__numeral,axiom,
! [M: num,N: num] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% minus_numeral_mod_numeral
thf(fact_6580_numeral__mod__minus__numeral,axiom,
! [M: num,N: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ) ).
% numeral_mod_minus_numeral
thf(fact_6581_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_6582_Divides_Oadjust__mod__def,axiom,
( adjust_mod
= ( ^ [L2: int,R5: int] :
( if @ int
@ ( R5
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ L2 @ R5 ) ) ) ) ).
% Divides.adjust_mod_def
thf(fact_6583_normalize__def,axiom,
( normalize
= ( ^ [P6: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P6 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P6 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_6584_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L2
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( sgn_sgn @ code_integer @ K3 )
= ( sgn_sgn @ code_integer @ L2 ) )
@ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S8: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S8
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S8 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_6585_gcd__right__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_gcd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ B2 )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% gcd_right_idem
thf(fact_6586_gcd__left__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ A2 @ B2 ) )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% gcd_left_idem
thf(fact_6587_gcd__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( ( gcd_gcd @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% gcd_eq_0_iff
thf(fact_6588_gcd_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] :
( ( gcd_gcd @ A @ ( one_one @ A ) @ A2 )
= ( one_one @ A ) ) ) ).
% gcd.bottom_left_bottom
thf(fact_6589_gcd_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] :
( ( gcd_gcd @ A @ A2 @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.bottom_right_bottom
thf(fact_6590_gcd__add1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,N: A] :
( ( gcd_gcd @ A @ ( plus_plus @ A @ M @ N ) @ N )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_add1
thf(fact_6591_gcd__add2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,N: A] :
( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ M @ N ) )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_add2
thf(fact_6592_gcd__neg2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_gcd @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% gcd_neg2
thf(fact_6593_gcd__neg1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% gcd_neg1
thf(fact_6594_gcd__exp,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( gcd_gcd @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( power_power @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ N ) ) ) ).
% gcd_exp
thf(fact_6595_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F2: C > B,G: A > C] :
( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F2 ) @ ( map @ A @ C @ G ) )
= ( map @ A @ B @ ( comp @ C @ B @ A @ F2 @ G ) ) ) ).
% map_comp_map
thf(fact_6596_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > C,G: A > B] :
( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F2 ) @ ( map @ A @ B @ G ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) ) ) ).
% List.map.comp
thf(fact_6597_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F2: A > B,V2: list @ A] :
( ( map @ B @ C @ G @ ( map @ A @ B @ F2 @ V2 ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ V2 ) ) ).
% list.map_comp
thf(fact_6598_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > C,G: A > B,List: list @ A] :
( ( map @ B @ C @ F2 @ ( map @ A @ B @ G @ List ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_6599_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F2: B > A,G: C > B,Xs: list @ C] :
( ( map @ B @ A @ F2 @ ( map @ C @ B @ G @ Xs ) )
= ( map @ C @ A @ ( comp @ B @ A @ C @ F2 @ G ) @ Xs ) ) ).
% map_map
thf(fact_6600_gcd__dvd1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ A2 ) ) ).
% gcd_dvd1
thf(fact_6601_gcd__dvd2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ B2 ) ) ).
% gcd_dvd2
thf(fact_6602_gcd__greatest__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) )
= ( ( dvd_dvd @ A @ A2 @ B2 )
& ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% gcd_greatest_iff
thf(fact_6603_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd @ int @ M @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% gcd_1_int
thf(fact_6604_gcd__neg2__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y ) )
= ( gcd_gcd @ int @ X @ Y ) ) ).
% gcd_neg2_int
thf(fact_6605_gcd__neg1__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y )
= ( gcd_gcd @ int @ X @ Y ) ) ).
% gcd_neg1_int
thf(fact_6606_abs__gcd__int,axiom,
! [X: int,Y: int] :
( ( abs_abs @ int @ ( gcd_gcd @ int @ X @ Y ) )
= ( gcd_gcd @ int @ X @ Y ) ) ).
% abs_gcd_int
thf(fact_6607_gcd__abs1__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd @ int @ ( abs_abs @ int @ X ) @ Y )
= ( gcd_gcd @ int @ X @ Y ) ) ).
% gcd_abs1_int
thf(fact_6608_gcd__abs2__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd @ int @ X @ ( abs_abs @ int @ Y ) )
= ( gcd_gcd @ int @ X @ Y ) ) ).
% gcd_abs2_int
thf(fact_6609_gcd__neg__numeral__1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: num,A2: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A2 )
= ( gcd_gcd @ A @ ( numeral_numeral @ A @ N ) @ A2 ) ) ) ).
% gcd_neg_numeral_1
thf(fact_6610_gcd__neg__numeral__2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,N: num] :
( ( gcd_gcd @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( gcd_gcd @ A @ A2 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% gcd_neg_numeral_2
thf(fact_6611_is__unit__gcd__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( one_one @ A ) )
= ( ( gcd_gcd @ A @ A2 @ B2 )
= ( one_one @ A ) ) ) ) ).
% is_unit_gcd_iff
thf(fact_6612_gcd__pos__int,axiom,
! [M: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ M @ N ) )
= ( ( M
!= ( zero_zero @ int ) )
| ( N
!= ( zero_zero @ int ) ) ) ) ).
% gcd_pos_int
thf(fact_6613_gcd__neg__numeral__1__int,axiom,
! [N: num,X: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ X )
= ( gcd_gcd @ int @ ( numeral_numeral @ int @ N ) @ X ) ) ).
% gcd_neg_numeral_1_int
thf(fact_6614_gcd__neg__numeral__2__int,axiom,
! [X: int,N: num] :
( ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( gcd_gcd @ int @ X @ ( numeral_numeral @ int @ N ) ) ) ).
% gcd_neg_numeral_2_int
thf(fact_6615_gcd__0__int,axiom,
! [X: int] :
( ( gcd_gcd @ int @ X @ ( zero_zero @ int ) )
= ( abs_abs @ int @ X ) ) ).
% gcd_0_int
thf(fact_6616_gcd__0__left__int,axiom,
! [X: int] :
( ( gcd_gcd @ int @ ( zero_zero @ int ) @ X )
= ( abs_abs @ int @ X ) ) ).
% gcd_0_left_int
thf(fact_6617_gcd__proj2__if__dvd__int,axiom,
! [Y: int,X: int] :
( ( dvd_dvd @ int @ Y @ X )
=> ( ( gcd_gcd @ int @ X @ Y )
= ( abs_abs @ int @ Y ) ) ) ).
% gcd_proj2_if_dvd_int
thf(fact_6618_gcd__proj1__if__dvd__int,axiom,
! [X: int,Y: int] :
( ( dvd_dvd @ int @ X @ Y )
=> ( ( gcd_gcd @ int @ X @ Y )
= ( abs_abs @ int @ X ) ) ) ).
% gcd_proj1_if_dvd_int
thf(fact_6619_foldr__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: B > A > A,F2: C > B,Xs: list @ C,A2: A] :
( ( foldr @ B @ A @ G @ ( map @ C @ B @ F2 @ Xs ) @ A2 )
= ( foldr @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F2 ) @ Xs @ A2 ) ) ).
% foldr_map
thf(fact_6620_foldr__Cons,axiom,
! [B: $tType,A: $tType,F2: A > B > B,X: A,Xs: list @ A] :
( ( foldr @ A @ B @ F2 @ ( cons @ A @ X @ Xs ) )
= ( comp @ B @ B @ B @ ( F2 @ X ) @ ( foldr @ A @ B @ F2 @ Xs ) ) ) ).
% foldr_Cons
thf(fact_6621_gcd__ge__0__int,axiom,
! [X: int,Y: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X @ Y ) ) ).
% gcd_ge_0_int
thf(fact_6622_gcd__red__int,axiom,
( ( gcd_gcd @ int )
= ( ^ [X2: int,Y4: int] : ( gcd_gcd @ int @ Y4 @ ( modulo_modulo @ int @ X2 @ Y4 ) ) ) ) ).
% gcd_red_int
thf(fact_6623_comp__cong,axiom,
! [C: $tType,B: $tType,D3: $tType,A: $tType,E4: $tType,F2: B > A,G: C > B,X: C,F7: D3 > A,G4: E4 > D3,X7: E4] :
( ( ( F2 @ ( G @ X ) )
= ( F7 @ ( G4 @ X7 ) ) )
=> ( ( comp @ B @ A @ C @ F2 @ G @ X )
= ( comp @ D3 @ A @ E4 @ F7 @ G4 @ X7 ) ) ) ).
% comp_cong
thf(fact_6624_gcd_Oleft__commute,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( gcd_gcd @ A @ B2 @ ( gcd_gcd @ A @ A2 @ C2 ) )
= ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd.left_commute
thf(fact_6625_gcd_Ocommute,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_gcd @ A )
= ( ^ [A5: A,B5: A] : ( gcd_gcd @ A @ B5 @ A5 ) ) ) ) ).
% gcd.commute
thf(fact_6626_gcd_Oassoc,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( gcd_gcd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 )
= ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd.assoc
thf(fact_6627_gcd__add__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,K: A,N: A] :
( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ ( times_times @ A @ K @ M ) @ N ) )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_add_mult
thf(fact_6628_gcd__diff1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [M: A,N: A] :
( ( gcd_gcd @ A @ ( minus_minus @ A @ M @ N ) @ N )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_diff1
thf(fact_6629_gcd__diff2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: A,M: A] :
( ( gcd_gcd @ A @ ( minus_minus @ A @ N @ M ) @ N )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_diff2
thf(fact_6630_bezout__int,axiom,
! [X: int,Y: int] :
? [U3: int,V4: int] :
( ( plus_plus @ int @ ( times_times @ int @ U3 @ X ) @ ( times_times @ int @ V4 @ Y ) )
= ( gcd_gcd @ int @ X @ Y ) ) ).
% bezout_int
thf(fact_6631_gcd__mult__distrib__int,axiom,
! [K: int,M: int,N: int] :
( ( times_times @ int @ ( abs_abs @ int @ K ) @ ( gcd_gcd @ int @ M @ N ) )
= ( gcd_gcd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N ) ) ) ).
% gcd_mult_distrib_int
thf(fact_6632_gcd__idem__int,axiom,
! [X: int] :
( ( gcd_gcd @ int @ X @ X )
= ( abs_abs @ int @ X ) ) ).
% gcd_idem_int
thf(fact_6633_gcd__dvd__prod,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,K: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( times_times @ A @ K @ B2 ) ) ) ).
% gcd_dvd_prod
thf(fact_6634_gcd__mono,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A,D4: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ D4 )
=> ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( gcd_gcd @ A @ C2 @ D4 ) ) ) ) ) ).
% gcd_mono
thf(fact_6635_dvd__gcdD1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) )
=> ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_gcdD1
thf(fact_6636_dvd__gcdD2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) )
=> ( dvd_dvd @ A @ A2 @ C2 ) ) ) ).
% dvd_gcdD2
thf(fact_6637_gcd__dvdI1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% gcd_dvdI1
thf(fact_6638_gcd__dvdI2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ C2 )
=> ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% gcd_dvdI2
thf(fact_6639_gcd__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ ( gcd_gcd @ A @ A2 @ B2 ) ) ) ) ) ).
% gcd_greatest
thf(fact_6640_gcd__dvd__antisym,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( gcd_gcd @ A @ C2 @ D4 ) )
=> ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ C2 @ D4 ) @ ( gcd_gcd @ A @ A2 @ B2 ) )
=> ( ( gcd_gcd @ A @ A2 @ B2 )
= ( gcd_gcd @ A @ C2 @ D4 ) ) ) ) ) ).
% gcd_dvd_antisym
thf(fact_6641_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H2: B > A,G: C > B,A3: set @ C] :
( ( ( H2 @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X3: B,Y3: B] :
( ( H2 @ ( plus_plus @ B @ X3 @ Y3 ) )
= ( plus_plus @ A @ ( H2 @ X3 ) @ ( H2 @ Y3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G ) @ A3 )
= ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A3 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_6642_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_6643_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_6644_gcd__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A2 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit2
thf(fact_6645_gcd__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit1
thf(fact_6646_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F7: A > B,A9: set @ A,A15: set @ B,F2: C > A,A3: set @ C] :
( ( bij_betw @ A @ B @ F7 @ A9 @ A15 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ A3 ) @ A9 )
=> ( ( bij_betw @ C @ A @ F2 @ A3 @ A9 )
= ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F7 @ F2 ) @ A3 @ A15 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_6647_gcd__le2__int,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ B2 ) ) ).
% gcd_le2_int
thf(fact_6648_gcd__le1__int,axiom,
! [A2: int,B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ A2 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ A2 ) ) ).
% gcd_le1_int
thf(fact_6649_uminus__sum__list__map,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( uminus_uminus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ ( comp @ A @ A @ B @ ( uminus_uminus @ A ) @ F2 ) @ Xs ) ) ) ) ).
% uminus_sum_list_map
thf(fact_6650_gcd__cases__int,axiom,
! [X: int,Y: int,P: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_gcd @ int @ X @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( P @ ( gcd_gcd @ int @ X @ Y ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_6651_gcd__unique__int,axiom,
! [D4: int,A2: int,B2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D4 )
& ( dvd_dvd @ int @ D4 @ A2 )
& ( dvd_dvd @ int @ D4 @ B2 )
& ! [E3: int] :
( ( ( dvd_dvd @ int @ E3 @ A2 )
& ( dvd_dvd @ int @ E3 @ B2 ) )
=> ( dvd_dvd @ int @ E3 @ D4 ) ) )
= ( D4
= ( gcd_gcd @ int @ A2 @ B2 ) ) ) ).
% gcd_unique_int
thf(fact_6652_gcd__non__0__int,axiom,
! [Y: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Y )
=> ( ( gcd_gcd @ int @ X @ Y )
= ( gcd_gcd @ int @ Y @ ( modulo_modulo @ int @ X @ Y ) ) ) ) ).
% gcd_non_0_int
thf(fact_6653_gcd__code__int,axiom,
( ( gcd_gcd @ int )
= ( ^ [K3: int,L2: int] :
( abs_abs @ int
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( gcd_gcd @ int @ L2 @ ( modulo_modulo @ int @ ( abs_abs @ int @ K3 ) @ ( abs_abs @ int @ L2 ) ) ) ) ) ) ) ).
% gcd_code_int
thf(fact_6654_case__prod__comp,axiom,
! [D3: $tType,A: $tType,C: $tType,B: $tType,F2: D3 > C > A,G: B > D3,X: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D3 @ ( C > A ) @ B @ F2 @ G ) @ X )
= ( F2 @ ( G @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% case_prod_comp
thf(fact_6655_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,H2: B > C,G: C > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X3: B,Y3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ( member @ B @ Y3 @ A3 )
=> ( ( X3 != Y3 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y3 ) )
=> ( ( G @ ( H2 @ X3 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image @ B @ C @ H2 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A3 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_6656_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% sum.eq_fold
thf(fact_6657_gcd__is__Max__divisors__int,axiom,
! [N: int,M: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( gcd_gcd @ int @ M @ N )
= ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D5: int] :
( ( dvd_dvd @ int @ D5 @ M )
& ( dvd_dvd @ int @ D5 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_int
thf(fact_6658_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I6: set @ C,G: A > B,F2: C > A] :
( ( finite_finite2 @ C @ I6 )
=> ( ! [I2: C] :
( ( member @ C @ I2 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F2 @ I2 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image @ C @ A @ F2 @ I6 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F2 ) @ I6 ) ) ) ) ) ).
% sum_image_le
thf(fact_6659_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_6660_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M3: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M3 @ K3 ) @ ( product_Pair @ nat @ nat @ M3 @ ( minus_minus @ nat @ K3 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M3 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_6661_gcd__nat_Oeq__neutr__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( gcd_gcd @ nat @ A2 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.eq_neutr_iff
thf(fact_6662_gcd__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% gcd_nat.left_neutral
thf(fact_6663_gcd__nat_Oneutr__eq__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( gcd_gcd @ nat @ A2 @ B2 ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.neutr_eq_iff
thf(fact_6664_gcd__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% gcd_nat.right_neutral
thf(fact_6665_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ X @ ( zero_zero @ nat ) )
= X ) ).
% gcd_0_nat
thf(fact_6666_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X )
= X ) ).
% gcd_0_left_nat
thf(fact_6667_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( one_one @ nat ) )
= ( one_one @ nat ) ) ).
% gcd_1_nat
thf(fact_6668_gcd__nat_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( gcd_gcd @ nat @ A2 @ B2 )
= A2 ) ) ).
% gcd_nat.absorb1
thf(fact_6669_gcd__nat_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( dvd_dvd @ nat @ B2 @ A2 )
=> ( ( gcd_gcd @ nat @ A2 @ B2 )
= B2 ) ) ).
% gcd_nat.absorb2
thf(fact_6670_gcd__nat_Obounded__iff,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) )
= ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ).
% gcd_nat.bounded_iff
thf(fact_6671_gcd__proj1__if__dvd__nat,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd @ nat @ X @ Y )
=> ( ( gcd_gcd @ nat @ X @ Y )
= X ) ) ).
% gcd_proj1_if_dvd_nat
thf(fact_6672_gcd__proj2__if__dvd__nat,axiom,
! [Y: nat,X: nat] :
( ( dvd_dvd @ nat @ Y @ X )
=> ( ( gcd_gcd @ nat @ X @ Y )
= Y ) ) ).
% gcd_proj2_if_dvd_nat
thf(fact_6673_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% gcd_Suc_0
thf(fact_6674_gcd__pos__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_gcd @ nat @ M @ N ) )
= ( ( M
!= ( zero_zero @ nat ) )
| ( N
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_pos_nat
thf(fact_6675_gcd__int__int__eq,axiom,
! [M: nat,N: nat] :
( ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ M @ N ) ) ) ).
% gcd_int_int_eq
thf(fact_6676_size__list__map,axiom,
! [A: $tType,B: $tType,F2: A > nat,G: B > A,Xs: list @ B] :
( ( size_list @ A @ F2 @ ( map @ B @ A @ G @ Xs ) )
= ( size_list @ B @ ( comp @ A @ nat @ B @ F2 @ G ) @ Xs ) ) ).
% size_list_map
thf(fact_6677_length__filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ ( map @ B @ A @ F2 @ Xs ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_6678_gcd__nat__abs__right__eq,axiom,
! [N: nat,K: int] :
( ( gcd_gcd @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( nat2 @ ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% gcd_nat_abs_right_eq
thf(fact_6679_gcd__nat__abs__left__eq,axiom,
! [K: int,N: nat] :
( ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( nat2 @ ( gcd_gcd @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ) ).
% gcd_nat_abs_left_eq
thf(fact_6680_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 )
@ ^ [X2: B] : ( product_Pair @ B @ B @ X2 @ X2 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_6681_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 )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_6682_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 )
@ ^ [X2: B] : ( product_Pair @ B @ B @ X2 @ X2 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_6683_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 )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_6684_filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] :
( ( filter2 @ A @ P @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ).
% filter_map
thf(fact_6685_list_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F2: B > nat,G: A > B] :
( ( comp @ ( list @ B ) @ nat @ ( list @ A ) @ ( size_list @ B @ F2 ) @ ( map @ A @ B @ G ) )
= ( size_list @ A @ ( comp @ B @ nat @ A @ F2 @ G ) ) ) ).
% list.size_gen_o_map
thf(fact_6686_takeWhile__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] :
( ( takeWhile @ A @ P @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( takeWhile @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ).
% takeWhile_map
thf(fact_6687_gcd__le1__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ A2 ) ) ).
% gcd_le1_nat
thf(fact_6688_gcd__le2__nat,axiom,
! [B2: nat,A2: nat] :
( ( B2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) ) ).
% gcd_le2_nat
thf(fact_6689_gcd__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( gcd_gcd @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( gcd_gcd @ int @ Xa @ X ) ) ) ).
% gcd_integer.abs_eq
thf(fact_6690_gcd__red__nat,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X2: nat,Y4: nat] : ( gcd_gcd @ nat @ Y4 @ ( modulo_modulo @ nat @ X2 @ Y4 ) ) ) ) ).
% gcd_red_nat
thf(fact_6691_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_6692_gcd__nat_Osimps,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X2: nat,Y4: nat] :
( if @ nat
@ ( Y4
= ( zero_zero @ nat ) )
@ X2
@ ( gcd_gcd @ nat @ Y4 @ ( modulo_modulo @ nat @ X2 @ Y4 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_6693_gcd__nat_Oelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa )
= Y )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y = X ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_6694_gcd__diff2__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ N @ M ) @ N )
= ( gcd_gcd @ nat @ M @ N ) ) ) ).
% gcd_diff2_nat
thf(fact_6695_gcd__diff1__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ M @ N ) @ N )
= ( gcd_gcd @ nat @ M @ N ) ) ) ).
% gcd_diff1_nat
thf(fact_6696_gcd__mult__distrib__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( gcd_gcd @ nat @ M @ N ) )
= ( gcd_gcd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% gcd_mult_distrib_nat
thf(fact_6697_gcd__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( gcd_gcd @ code_integer @ X @ Xa ) )
= ( gcd_gcd @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% gcd_integer.rep_eq
thf(fact_6698_gcd__nat_Omono,axiom,
! [A2: nat,C2: nat,B2: nat,D4: nat] :
( ( dvd_dvd @ nat @ A2 @ C2 )
=> ( ( dvd_dvd @ nat @ B2 @ D4 )
=> ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ ( gcd_gcd @ nat @ C2 @ D4 ) ) ) ) ).
% gcd_nat.mono
thf(fact_6699_gcd__nat_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( A2
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ).
% gcd_nat.orderE
thf(fact_6700_gcd__nat_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( gcd_gcd @ nat @ A2 @ B2 ) )
=> ( dvd_dvd @ nat @ A2 @ B2 ) ) ).
% gcd_nat.orderI
thf(fact_6701_gcd__nat_Oabsorb3,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( ( gcd_gcd @ nat @ A2 @ B2 )
= A2 ) ) ).
% gcd_nat.absorb3
thf(fact_6702_gcd__nat_Oabsorb4,axiom,
! [B2: nat,A2: nat] :
( ( ( dvd_dvd @ nat @ B2 @ A2 )
& ( B2 != A2 ) )
=> ( ( gcd_gcd @ nat @ A2 @ B2 )
= B2 ) ) ).
% gcd_nat.absorb4
thf(fact_6703_gcd__nat_OboundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) )
=> ~ ( ( dvd_dvd @ nat @ A2 @ B2 )
=> ~ ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ).
% gcd_nat.boundedE
thf(fact_6704_gcd__nat_OboundedI,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
=> ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) ) ) ) ).
% gcd_nat.boundedI
thf(fact_6705_gcd__nat_Oorder__iff,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A5: nat,B5: nat] :
( A5
= ( gcd_gcd @ nat @ A5 @ B5 ) ) ) ) ).
% gcd_nat.order_iff
thf(fact_6706_gcd__nat_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ A2 ) ).
% gcd_nat.cobounded1
thf(fact_6707_gcd__nat_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) ).
% gcd_nat.cobounded2
thf(fact_6708_gcd__nat_Oabsorb__iff1,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A5: nat,B5: nat] :
( ( gcd_gcd @ nat @ A5 @ B5 )
= A5 ) ) ) ).
% gcd_nat.absorb_iff1
thf(fact_6709_gcd__nat_Oabsorb__iff2,axiom,
( ( dvd_dvd @ nat )
= ( ^ [B5: nat,A5: nat] :
( ( gcd_gcd @ nat @ A5 @ B5 )
= B5 ) ) ) ).
% gcd_nat.absorb_iff2
thf(fact_6710_gcd__nat_OcoboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A2 @ C2 )
=> ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) ) ).
% gcd_nat.coboundedI1
thf(fact_6711_gcd__nat_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) ) ).
% gcd_nat.coboundedI2
thf(fact_6712_gcd__nat_Ostrict__boundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) )
& ( A2
!= ( gcd_gcd @ nat @ B2 @ C2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ~ ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_boundedE
thf(fact_6713_gcd__nat_Ostrict__order__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
= ( ( A2
= ( gcd_gcd @ nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ).
% gcd_nat.strict_order_iff
thf(fact_6714_gcd__nat_Ostrict__coboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) )
=> ( ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 )
& ( ( gcd_gcd @ nat @ A2 @ B2 )
!= C2 ) ) ) ).
% gcd_nat.strict_coboundedI1
thf(fact_6715_gcd__nat_Ostrict__coboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 )
& ( ( gcd_gcd @ nat @ A2 @ B2 )
!= C2 ) ) ) ).
% gcd_nat.strict_coboundedI2
thf(fact_6716_gcd__unique__nat,axiom,
! [D4: nat,A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ D4 @ A2 )
& ( dvd_dvd @ nat @ D4 @ B2 )
& ! [E3: nat] :
( ( ( dvd_dvd @ nat @ E3 @ A2 )
& ( dvd_dvd @ nat @ E3 @ B2 ) )
=> ( dvd_dvd @ nat @ E3 @ D4 ) ) )
= ( D4
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ).
% gcd_unique_nat
thf(fact_6717_folding__insort__key_Oinsort__key__commute,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F2: B > A,X: B,Y: B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F2 )
=> ( ( member @ B @ X @ S2 )
=> ( ( member @ B @ Y @ S2 )
=> ( ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq @ F2 @ Y ) @ ( insort_key @ A @ B @ Less_eq @ F2 @ X ) )
= ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq @ F2 @ X ) @ ( insort_key @ A @ B @ Less_eq @ F2 @ Y ) ) ) ) ) ) ).
% folding_insort_key.insort_key_commute
thf(fact_6718_bezout__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [X3: nat,Y3: nat] :
( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_nat
thf(fact_6719_bezout__gcd__nat_H,axiom,
! [B2: nat,A2: nat] :
? [X3: nat,Y3: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ ( times_times @ nat @ A2 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X3 ) @ ( times_times @ nat @ B2 @ Y3 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A2 @ Y3 ) @ ( times_times @ nat @ B2 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A2 @ Y3 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_6720_bit__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( plus_plus @ nat @ N ) ) ) ) ).
% bit_drop_bit_eq
thf(fact_6721_summable__inverse__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ ( comp @ A @ A @ nat @ ( inverse_inverse @ A ) @ F2 ) )
=> ( summable @ A
@ ^ [N3: nat] : ( divide_divide @ A @ C2 @ ( F2 @ N3 ) ) ) ) ) ).
% summable_inverse_divide
thf(fact_6722_gcd__code__integer,axiom,
( ( gcd_gcd @ code_integer )
= ( ^ [K3: code_integer,L2: code_integer] :
( abs_abs @ code_integer
@ ( if @ code_integer
@ ( L2
= ( zero_zero @ code_integer ) )
@ K3
@ ( gcd_gcd @ code_integer @ L2 @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ) ) ).
% gcd_code_integer
thf(fact_6723_num__of__nat__code,axiom,
( num_of_nat
= ( comp @ code_integer @ num @ nat @ code_num_of_integer @ ( semiring_1_of_nat @ code_integer ) ) ) ).
% num_of_nat_code
thf(fact_6724_gcd__int__def,axiom,
( ( gcd_gcd @ int )
= ( ^ [X2: int,Y4: int] : ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ X2 ) ) @ ( nat2 @ ( abs_abs @ int @ Y4 ) ) ) ) ) ) ).
% gcd_int_def
thf(fact_6725_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( gcd_gcd @ nat @ M @ N )
= ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D5: nat] :
( ( dvd_dvd @ nat @ D5 @ M )
& ( dvd_dvd @ nat @ D5 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_6726_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_6727_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_6728_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_6729_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_6730_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_6731_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_6732_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_6733_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_6734_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_6735_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_6736_sum_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_int_atMost_int_shift
thf(fact_6737_prod_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_int_atMost_int_shift
thf(fact_6738_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_6739_sum_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_int_lessThan_int_shift
thf(fact_6740_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F3: A > ( option @ B ),Xs2: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F3 )
@ ( filter2 @ A
@ ^ [X2: A] :
( ( F3 @ X2 )
!= ( none @ B ) )
@ Xs2 ) ) ) ) ).
% map_filter_def
thf(fact_6741_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_6742_prod_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_int_lessThan_int_shift
thf(fact_6743_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_6744_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_6745_gcd__nat_Opelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y = X ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_6746_gcd__idem__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ X @ X )
= X ) ).
% gcd_idem_nat
thf(fact_6747_gcd__nat_Oright__idem,axiom,
! [A2: nat,B2: nat] :
( ( gcd_gcd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ).
% gcd_nat.right_idem
thf(fact_6748_gcd__nat_Oleft__idem,axiom,
! [A2: nat,B2: nat] :
( ( gcd_gcd @ nat @ A2 @ ( gcd_gcd @ nat @ A2 @ B2 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ).
% gcd_nat.left_idem
thf(fact_6749_gcd__nat_Oidem,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ A2 @ A2 )
= A2 ) ).
% gcd_nat.idem
thf(fact_6750_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 )
@ ^ [X2: B,Y4: A] : ( product_Pair @ A @ B @ Y4 @ X2 ) ) ) ) ).
% snd_fst_flip
thf(fact_6751_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 )
@ ^ [X2: A,Y4: B] : ( product_Pair @ B @ A @ Y4 @ X2 ) ) ) ) ).
% fst_snd_flip
thf(fact_6752_refl__ge__eq,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X3: A] : ( R2 @ X3 @ X3 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ R2 ) ) ).
% refl_ge_eq
thf(fact_6753_ge__eq__refl,axiom,
! [A: $tType,R2: A > A > $o,X: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ R2 )
=> ( R2 @ X @ X ) ) ).
% ge_eq_refl
thf(fact_6754_Code__Numeral_Onegative__def,axiom,
( code_negative
= ( comp @ code_integer @ code_integer @ num @ ( uminus_uminus @ code_integer ) @ ( numeral_numeral @ code_integer ) ) ) ).
% Code_Numeral.negative_def
thf(fact_6755_Code__Target__Int_Onegative__def,axiom,
( code_Target_negative
= ( comp @ int @ int @ num @ ( uminus_uminus @ int ) @ ( numeral_numeral @ int ) ) ) ).
% Code_Target_Int.negative_def
thf(fact_6756_empty__natural,axiom,
! [C: $tType,B: $tType,D3: $tType,A: $tType,F2: A > C,G: D3 > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu: C] : ( bot_bot @ ( set @ B ) )
@ F2 )
= ( comp @ ( set @ D3 ) @ ( set @ B ) @ A @ ( image @ D3 @ B @ G )
@ ^ [Uu: A] : ( bot_bot @ ( set @ D3 ) ) ) ) ).
% empty_natural
thf(fact_6757_finite__enumerate,axiom,
! [S2: set @ nat] :
( ( finite_finite2 @ nat @ S2 )
=> ? [R: nat > nat] :
( ( strict_mono_on @ nat @ nat @ R @ ( set_ord_lessThan @ nat @ ( finite_card @ nat @ S2 ) ) )
& ! [N9: nat] :
( ( ord_less @ nat @ N9 @ ( finite_card @ nat @ S2 ) )
=> ( member @ nat @ ( R @ N9 ) @ S2 ) ) ) ) ).
% finite_enumerate
thf(fact_6758_strict__mono__on__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F2: A > B,A3: set @ A,X: A,Y: A] :
( ( strict_mono_on @ A @ B @ F2 @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_6759_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F2: A > B,A3: set @ A,R3: A,S: A] :
( ( strict_mono_on @ A @ B @ F2 @ A3 )
=> ( ( member @ A @ R3 @ A3 )
=> ( ( member @ A @ S @ A3 )
=> ( ( ord_less @ A @ R3 @ S )
=> ( ord_less @ B @ ( F2 @ R3 ) @ ( F2 @ S ) ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_6760_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A3: set @ A,F2: A > B] :
( ! [R: A,S3: A] :
( ( member @ A @ R @ A3 )
=> ( ( member @ A @ S3 @ A3 )
=> ( ( ord_less @ A @ R @ S3 )
=> ( ord_less @ B @ ( F2 @ R ) @ ( F2 @ S3 ) ) ) ) )
=> ( strict_mono_on @ A @ B @ F2 @ A3 ) ) ) ).
% strict_mono_onI
thf(fact_6761_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( strict_mono_on @ A @ B )
= ( ^ [F3: A > B,A4: set @ A] :
! [R5: A,S8: A] :
( ( ( member @ A @ R5 @ A4 )
& ( member @ A @ S8 @ A4 )
& ( ord_less @ A @ R5 @ S8 ) )
=> ( ord_less @ B @ ( F3 @ R5 ) @ ( F3 @ S8 ) ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_6762_conj__subset__def,axiom,
! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A3
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_6763_conj__comp__iff,axiom,
! [B: $tType,A: $tType,P: B > $o,Q: B > $o,G: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X2: B] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ G )
= ( ^ [X2: A] :
( ( comp @ B @ $o @ A @ P @ G @ X2 )
& ( comp @ B @ $o @ A @ Q @ G @ X2 ) ) ) ) ).
% conj_comp_iff
thf(fact_6764_card__UNION,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A3 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A3 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) )
= ( 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 @ A3 )
& ( I7
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_6765_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ ( finite_Fpow @ B @ A3 ) ) @ ( finite_Fpow @ A @ B3 ) ) ) ).
% image_Fpow_mono
thf(fact_6766_cSup__lessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A )
& ( no_bot @ A ) )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_lessThan @ A @ X ) )
= X ) ) ).
% cSup_lessThan
thf(fact_6767_cSup__atMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_atMost @ A @ X ) )
= X ) ) ).
% cSup_atMost
thf(fact_6768_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_6769_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_6770_cSup__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cSup_singleton
thf(fact_6771_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_6772_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_6773_cInf__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cInf_singleton
thf(fact_6774_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastLessThan
thf(fact_6775_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_6776_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastLessThan
thf(fact_6777_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_6778_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X2: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% cSUP_const
thf(fact_6779_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X2: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% cINF_const
thf(fact_6780_finite__UN__I,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( finite_finite2 @ B @ ( B3 @ A6 ) ) )
=> ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ) ).
% finite_UN_I
thf(fact_6781_finite__INT,axiom,
! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] :
( ? [X6: A] :
( ( member @ A @ X6 @ I6 )
& ( finite_finite2 @ B @ ( A3 @ X6 ) ) )
=> ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) ) ).
% finite_INT
thf(fact_6782_Union__natural,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( comp @ ( set @ ( set @ B ) ) @ ( set @ B ) @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ B ) ) @ ( image @ ( set @ A ) @ ( set @ B ) @ ( image @ A @ B @ F2 ) ) )
= ( comp @ ( set @ A ) @ ( set @ B ) @ ( set @ ( set @ A ) ) @ ( image @ A @ B @ F2 ) @ ( complete_Sup_Sup @ ( set @ A ) ) ) ) ).
% Union_natural
thf(fact_6783_in__Union__o__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,X: A,Gset: B > ( set @ ( set @ A ) ),Gmap: C > B,A3: C] :
( ( member @ A @ X @ ( comp @ B @ ( set @ A ) @ C @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ B @ ( complete_Sup_Sup @ ( set @ A ) ) @ Gset ) @ Gmap @ A3 ) )
=> ( member @ A @ X @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( comp @ B @ ( set @ ( set @ A ) ) @ C @ Gset @ Gmap ) @ A3 ) ) ) ).
% in_Union_o_assoc
thf(fact_6784_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F2: B > ( set @ A ),G: C > ( set @ B ),X: C,X5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ ( G @ X ) ) ) @ X5 )
= ( ord_less_eq @ ( set @ B ) @ ( G @ X )
@ ( collect @ B
@ ^ [X2: B] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ X2 ) @ X5 ) ) ) ) ).
% UN_image_subset
thf(fact_6785_finite__imp__less__Inf,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,X: A,A2: A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( member @ A @ X @ X5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ( ord_less @ A @ A2 @ X3 ) )
=> ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X5 ) ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_6786_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,X: A,A2: A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( member @ A @ X @ X5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ( ord_less @ A @ X3 @ A2 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X5 ) @ A2 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_6787_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,Z3: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X5 ) @ Z3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ X5 )
& ( ord_less @ A @ X3 @ Z3 ) ) ) ) ) ).
% cInf_lessD
thf(fact_6788_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y: A,X5: set @ A] :
( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X5 ) )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ) ) ).
% less_cSupE
thf(fact_6789_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,Z3: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z3 @ ( complete_Sup_Sup @ A @ X5 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ X5 )
& ( ord_less @ A @ Z3 @ X3 ) ) ) ) ) ).
% less_cSupD
thf(fact_6790_SUP__inf__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,A3: set @ B,G: C > A,B3: set @ C] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [A5: B] :
( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [B5: C] : ( inf_inf @ A @ ( F2 @ A5 ) @ ( G @ B5 ) )
@ B3 ) )
@ A3 ) ) ) ) ).
% SUP_inf_distrib2
thf(fact_6791_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A2: A,F2: B > A,B3: set @ B] :
( ( inf_inf @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ B3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [B5: B] : ( inf_inf @ A @ A2 @ ( F2 @ B5 ) )
@ B3 ) ) ) ) ).
% inf_SUP
thf(fact_6792_Sup__inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B3: set @ A,A2: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B3 ) @ A2 )
= ( complete_Sup_Sup @ A
@ ( image @ A @ A
@ ^ [B5: A] : ( inf_inf @ A @ B5 @ A2 )
@ B3 ) ) ) ) ).
% Sup_inf
thf(fact_6793_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,B3: set @ B,A2: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ B3 ) ) @ A2 )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [B5: B] : ( inf_inf @ A @ ( F2 @ B5 ) @ A2 )
@ B3 ) ) ) ) ).
% SUP_inf
thf(fact_6794_INF__sup__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,A3: set @ B,G: C > A,B3: set @ C] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B3 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [A5: B] :
( complete_Inf_Inf @ A
@ ( image @ C @ A
@ ^ [B5: C] : ( sup_sup @ A @ ( F2 @ A5 ) @ ( G @ B5 ) )
@ B3 ) )
@ A3 ) ) ) ) ).
% INF_sup_distrib2
thf(fact_6795_sup__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A2: A,F2: B > A,B3: set @ B] :
( ( sup_sup @ A @ A2 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ B3 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [B5: B] : ( sup_sup @ A @ A2 @ ( F2 @ B5 ) )
@ B3 ) ) ) ) ).
% sup_INF
thf(fact_6796_Inf__sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B3: set @ A,A2: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B3 ) @ A2 )
= ( complete_Inf_Inf @ A
@ ( image @ A @ A
@ ^ [B5: A] : ( sup_sup @ A @ B5 @ A2 )
@ B3 ) ) ) ) ).
% Inf_sup
thf(fact_6797_INF__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,B3: set @ B,A2: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ B3 ) ) @ A2 )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [B5: B] : ( sup_sup @ A @ ( F2 @ B5 ) @ A2 )
@ B3 ) ) ) ) ).
% INF_sup
thf(fact_6798_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,X: A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( member @ A @ X @ X5 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X5 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_6799_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z3: A,X5: set @ A] :
( ( member @ A @ Z3 @ X5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ X5 )
= Z3 ) ) ) ) ).
% cInf_eq_minimum
thf(fact_6800_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X5: set @ A,A2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ( ord_less_eq @ A @ A2 @ X3 ) )
=> ( ! [Y3: A] :
( ! [X6: A] :
( ( member @ A @ X6 @ X5 )
=> ( ord_less_eq @ A @ Y3 @ X6 ) )
=> ( ord_less_eq @ A @ Y3 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X5 )
= A2 ) ) ) ) ).
% cInf_eq
thf(fact_6801_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z3: A,X5: set @ A] :
( ( member @ A @ Z3 @ X5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ( complete_Sup_Sup @ A @ X5 )
= Z3 ) ) ) ) ).
% cSup_eq_maximum
thf(fact_6802_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X5: set @ A,A2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ( ord_less_eq @ A @ X3 @ A2 ) )
=> ( ! [Y3: A] :
( ! [X6: A] :
( ( member @ A @ X6 @ X5 )
=> ( ord_less_eq @ A @ X6 @ Y3 ) )
=> ( ord_less_eq @ A @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X5 )
= A2 ) ) ) ) ).
% cSup_eq
thf(fact_6803_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,X: A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( member @ A @ X @ X5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X5 ) @ X ) ) ) ) ).
% cInf_le_finite
thf(fact_6804_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,Z3: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ X5 ) ) ) ) ) ).
% cInf_greatest
thf(fact_6805_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A2: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ( ord_less_eq @ A @ A2 @ X3 ) )
=> ( ! [Y3: A] :
( ! [X6: A] :
( ( member @ A @ X6 @ X5 )
=> ( ord_less_eq @ A @ Y3 @ X6 ) )
=> ( ord_less_eq @ A @ Y3 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X5 )
= A2 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_6806_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,Z3: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X5 ) @ Z3 ) ) ) ) ).
% cSup_least
thf(fact_6807_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A2: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ( ord_less_eq @ A @ X3 @ A2 ) )
=> ( ! [Y3: A] :
( ! [X6: A] :
( ( member @ A @ X6 @ X5 )
=> ( ord_less_eq @ A @ X6 @ Y3 ) )
=> ( ord_less_eq @ A @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X5 )
= A2 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_6808_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_6809_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,M: A,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ M @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_6810_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F2: B > A,M7: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ M7 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ M7 ) ) ) ) ).
% cSUP_least
thf(fact_6811_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,A2: A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X5 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X5 )
=> ( ord_less @ A @ A2 @ X2 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_6812_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,A2: A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X5 ) @ A2 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X5 )
=> ( ord_less @ A @ X2 @ A2 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_6813_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S2 ) ) @ A2 ) ) ) ) ).
% cInf_abs_ge
thf(fact_6814_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S2 ) ) @ A2 ) ) ) ) ).
% cSup_abs_le
thf(fact_6815_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: set @ ( set @ B ),G: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B3 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [A16: set @ B] :
( ( member @ ( set @ B ) @ A16 @ B3 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B3 )
=> ( ( A16 != A25 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A16 )
=> ( ( member @ B @ X3 @ A25 )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B3 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ B3 ) ) ) ) ) ).
% sum.Union_comp
thf(fact_6816_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Max_Sup
thf(fact_6817_cSup__eq__Max,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X5 )
= ( lattic643756798349783984er_Max @ A @ X5 ) ) ) ) ) ).
% cSup_eq_Max
thf(fact_6818_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ).
% Min_Inf
thf(fact_6819_cInf__eq__Min,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X5 )
= ( lattic643756798350308766er_Min @ A @ X5 ) ) ) ) ) ).
% cInf_eq_Min
thf(fact_6820_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_6821_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X5 )
= ( lattic7752659483105999362nf_fin @ A @ X5 ) ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_6822_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_6823_Union__image__insert,axiom,
! [A: $tType,B: $tType,F2: B > ( set @ A ),A2: B,B3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ ( insert @ B @ A2 @ B3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( F2 @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ B3 ) ) ) ) ).
% Union_image_insert
thf(fact_6824_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X5 )
= ( lattic5882676163264333800up_fin @ A @ X5 ) ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_6825_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_6826_UN__le__add__shift__strict,axiom,
! [A: $tType,M7: nat > ( set @ A ),K: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [I4: nat] : ( M7 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_6827_UN__le__add__shift,axiom,
! [A: $tType,M7: nat > ( set @ A ),K: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [I4: nat] : ( M7 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_6828_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,L: A,E2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S2 ) @ L ) ) @ E2 ) ) ) ) ).
% cInf_asclose
thf(fact_6829_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,L: A,E2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S2 ) @ L ) ) @ E2 ) ) ) ) ).
% cSup_asclose
thf(fact_6830_Fpow__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) @ ( finite_Fpow @ A @ B3 ) ) ) ).
% Fpow_mono
thf(fact_6831_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S2: set @ A,X: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ( S2
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ S2 ) )
= X ) )
& ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ S2 ) )
= ( ord_max @ A @ X @ ( complete_Sup_Sup @ A @ S2 ) ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_6832_Fpow__subset__Pow,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) @ ( pow2 @ A @ A3 ) ) ).
% Fpow_subset_Pow
thf(fact_6833_Fpow__def,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A4: set @ A] :
( collect @ ( set @ A )
@ ^ [X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ A4 )
& ( finite_finite2 @ A @ X8 ) ) ) ) ) ).
% Fpow_def
thf(fact_6834_finite__subset__Union,axiom,
! [A: $tType,A3: set @ A,B14: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B14 ) )
=> ~ ! [F8: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F8 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F8 @ B14 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ F8 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_6835_fold__union__pair,axiom,
! [B: $tType,A: $tType,B3: set @ A,X: B,A3: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B3 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y4: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B3 ) )
@ A3 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y4: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y4 ) )
@ A3
@ B3 ) ) ) ).
% fold_union_pair
thf(fact_6836_dvd__partition,axiom,
! [A: $tType,C3: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C3 )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X3 ) ) )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C3 )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ C3 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_6837_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,A3: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I6 )
=> ( finite_finite2 @ C @ ( A3 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I6 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I6 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X3 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ A3 @ I6 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( groups7311177749621191930dd_sum @ C @ A @ G @ ( A3 @ X2 ) )
@ I6 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_6838_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,A3: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I6 )
=> ( finite_finite2 @ C @ ( A3 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I6 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I6 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X3 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ A3 @ I6 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( groups7121269368397514597t_prod @ C @ A @ G @ ( A3 @ X2 ) )
@ I6 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_6839_UN__le__eq__Un0,axiom,
! [A: $tType,M7: nat > ( set @ A ),N: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_ord_atMost @ nat @ N ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M7 @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_6840_card__UN__le,axiom,
! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A3 @ I4 ) )
@ I6 ) ) ) ).
% card_UN_le
thf(fact_6841_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ I6 )
=> ( finite_finite2 @ B @ ( A3 @ X3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ I6 )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ I6 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ ( A3 @ X3 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A3 @ I4 ) )
@ I6 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_6842_Fpow__Pow__finite,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A4: set @ A] : ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A4 ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_6843_Compl__INT,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X2: B] : ( uminus_uminus @ ( set @ A ) @ ( B3 @ X2 ) )
@ A3 ) ) ) ).
% Compl_INT
thf(fact_6844_Compl__UN,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X2: B] : ( uminus_uminus @ ( set @ A ) @ ( B3 @ X2 ) )
@ A3 ) ) ) ).
% Compl_UN
thf(fact_6845_Sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A4: set @ ( A > B ),X2: A] :
( complete_Sup_Sup @ B
@ ( image @ ( A > B ) @ B
@ ^ [F3: A > B] : ( F3 @ X2 )
@ A4 ) ) ) ) ) ).
% Sup_apply
thf(fact_6846_Inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A4: set @ ( A > B ),X2: A] :
( complete_Inf_Inf @ B
@ ( image @ ( A > B ) @ B
@ ^ [F3: A > B] : ( F3 @ X2 )
@ A4 ) ) ) ) ) ).
% Inf_apply
thf(fact_6847_SUP__bool__eq,axiom,
! [A: $tType] :
( ( ^ [A4: set @ A,F3: A > $o] : ( complete_Sup_Sup @ $o @ ( image @ A @ $o @ F3 @ A4 ) ) )
= ( bex @ A ) ) ).
% SUP_bool_eq
thf(fact_6848_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_6849_SUP__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Sup @ A )
=> ! [F2: C > B > A,A3: set @ C,X: B] :
( ( complete_Sup_Sup @ ( B > A ) @ ( image @ C @ ( B > A ) @ F2 @ A3 ) @ X )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [Y4: C] : ( F2 @ Y4 @ X )
@ A3 ) ) ) ) ).
% SUP_apply
thf(fact_6850_SUP__identity__eq,axiom,
! [A: $tType] :
( ( complete_Sup @ A )
=> ! [A3: set @ A] :
( ( complete_Sup_Sup @ A
@ ( image @ A @ A
@ ^ [X2: A] : X2
@ A3 ) )
= ( complete_Sup_Sup @ A @ A3 ) ) ) ).
% SUP_identity_eq
thf(fact_6851_INF__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Inf @ A )
=> ! [F2: C > B > A,A3: set @ C,X: B] :
( ( complete_Inf_Inf @ ( B > A ) @ ( image @ C @ ( B > A ) @ F2 @ A3 ) @ X )
= ( complete_Inf_Inf @ A
@ ( image @ C @ A
@ ^ [Y4: C] : ( F2 @ Y4 @ X )
@ A3 ) ) ) ) ).
% INF_apply
thf(fact_6852_INF__identity__eq,axiom,
! [A: $tType] :
( ( complete_Inf @ A )
=> ! [A3: set @ A] :
( ( complete_Inf_Inf @ A
@ ( image @ A @ A
@ ^ [X2: A] : X2
@ A3 ) )
= ( complete_Inf_Inf @ A @ A3 ) ) ) ).
% INF_identity_eq
thf(fact_6853_UN__iff,axiom,
! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A3: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( member @ A @ B2 @ ( B3 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_6854_UN__I,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,B3: A > ( set @ B )] :
( ( member @ A @ A2 @ A3 )
=> ( ( member @ B @ B2 @ ( B3 @ A2 ) )
=> ( member @ B @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_6855_INT__iff,axiom,
! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A3: set @ B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( member @ A @ B2 @ ( B3 @ X2 ) ) ) ) ) ).
% INT_iff
thf(fact_6856_INT__I,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: B,B3: A > ( set @ B )] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ B @ B2 @ ( B3 @ X3 ) ) )
=> ( member @ B @ B2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ).
% INT_I
thf(fact_6857_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( ( complete_Inf_Inf @ A @ A3 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A3 )
& ( ord_less @ A @ Y4 @ X2 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_6858_Sup__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: A,A3: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ A3 ) )
= ( sup_sup @ A @ A2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Sup_insert
thf(fact_6859_Inf__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A2: A,A3: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ A3 ) )
= ( inf_inf @ A @ A2 @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ).
% Inf_insert
thf(fact_6860_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( bot_bot @ A )
@ A3 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_6861_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: B > A,A3: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ B3 @ A3 ) )
= ( bot_bot @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B3 @ X2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_6862_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: B > A,A3: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ B3 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B3 @ X2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_6863_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I4: B] : F2
@ A3 ) )
= F2 ) ) ) ).
% SUP_const
thf(fact_6864_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I4: B] : F2
@ A3 ) )
= F2 ) ) ) ).
% INF_const
thf(fact_6865_UN__constant,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y4: B] : C2
@ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y4: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_6866_UN__Un,axiom,
! [A: $tType,B: $tType,M7: B > ( set @ A ),A3: set @ B,B3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ M7 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ M7 @ A3 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ M7 @ B3 ) ) ) ) ).
% UN_Un
thf(fact_6867_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
=> ? [Y4: B] :
( ( member @ B @ Y4 @ A3 )
& ( ord_less @ A @ ( F2 @ Y4 ) @ X2 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_6868_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C3: set @ B,A2: A,B3: B > ( set @ A )] :
( ( ( C3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X2: B] : ( insert @ A @ A2 @ ( B3 @ X2 ) )
@ C3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X2: B] : ( insert @ A @ A2 @ ( B3 @ X2 ) )
@ C3 ) )
= ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_6869_UN__singleton,axiom,
! [A: $tType,A3: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ A @ ( set @ A )
@ ^ [X2: A] : ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) )
= A3 ) ).
% UN_singleton
thf(fact_6870_UN__simps_I3_J,axiom,
! [E4: $tType,F: $tType,C3: set @ F,A3: set @ E4,B3: F > ( set @ E4 )] :
( ( ( C3
= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E4 )
@ ( image @ F @ ( set @ E4 )
@ ^ [X2: F] : ( sup_sup @ ( set @ E4 ) @ A3 @ ( B3 @ X2 ) )
@ C3 ) )
= ( bot_bot @ ( set @ E4 ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E4 )
@ ( image @ F @ ( set @ E4 )
@ ^ [X2: F] : ( sup_sup @ ( set @ E4 ) @ A3 @ ( B3 @ X2 ) )
@ C3 ) )
= ( sup_sup @ ( set @ E4 ) @ A3 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image @ F @ ( set @ E4 ) @ B3 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_6871_UN__simps_I2_J,axiom,
! [C: $tType,D3: $tType,C3: set @ C,A3: C > ( set @ D3 ),B3: set @ D3] :
( ( ( C3
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D3 )
@ ( image @ C @ ( set @ D3 )
@ ^ [X2: C] : ( sup_sup @ ( set @ D3 ) @ ( A3 @ X2 ) @ B3 )
@ C3 ) )
= ( bot_bot @ ( set @ D3 ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D3 )
@ ( image @ C @ ( set @ D3 )
@ ^ [X2: C] : ( sup_sup @ ( set @ D3 ) @ ( A3 @ X2 ) @ B3 )
@ C3 ) )
= ( sup_sup @ ( set @ D3 ) @ ( complete_Sup_Sup @ ( set @ D3 ) @ ( image @ C @ ( set @ D3 ) @ A3 @ C3 ) ) @ B3 ) ) ) ) ).
% UN_simps(2)
thf(fact_6872_UN__insert,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A2: B,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ ( insert @ B @ A2 @ A3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( B3 @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% UN_insert
thf(fact_6873_INT__insert,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A2: B,A3: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ ( insert @ B @ A2 @ A3 ) ) )
= ( inf_inf @ ( set @ A ) @ ( B3 @ A2 ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% INT_insert
thf(fact_6874_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S6: set @ ( A > B > $o ),X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S6 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_6875_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S6: set @ ( A > B > $o ),X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S6 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_6876_Inf__int__def,axiom,
( ( complete_Inf_Inf @ int )
= ( ^ [X8: set @ int] : ( uminus_uminus @ int @ ( complete_Sup_Sup @ int @ ( image @ int @ int @ ( uminus_uminus @ int ) @ X8 ) ) ) ) ) ).
% Inf_int_def
thf(fact_6877_Inf__real__def,axiom,
( ( complete_Inf_Inf @ real )
= ( ^ [X8: set @ real] : ( uminus_uminus @ real @ ( complete_Sup_Sup @ real @ ( image @ real @ real @ ( uminus_uminus @ real ) @ X8 ) ) ) ) ) ).
% Inf_real_def
thf(fact_6878_Inf__nat__def1,axiom,
! [K6: set @ nat] :
( ( K6
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( complete_Inf_Inf @ nat @ K6 ) @ K6 ) ) ).
% Inf_nat_def1
thf(fact_6879_Inf__nat__def,axiom,
( ( complete_Inf_Inf @ nat )
= ( ^ [X8: set @ nat] :
( ord_Least @ nat
@ ^ [N3: nat] : ( member @ nat @ N3 @ X8 ) ) ) ) ).
% Inf_nat_def
thf(fact_6880_SUP__Sup__eq,axiom,
! [A: $tType,S2: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image @ ( set @ A ) @ ( A > $o )
@ ^ [I4: set @ A,X2: A] : ( member @ A @ X2 @ I4 )
@ S2 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_6881_INF__Int__eq,axiom,
! [A: $tType,S2: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image @ ( set @ A ) @ ( A > $o )
@ ^ [I4: set @ A,X2: A] : ( member @ A @ X2 @ I4 )
@ S2 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Inf_Inf @ ( set @ A ) @ S2 ) ) ) ) ).
% INF_Int_eq
thf(fact_6882_INF__INT__eq,axiom,
! [B: $tType,A: $tType,R3: B > ( set @ A ),S2: set @ B] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image @ B @ ( A > $o )
@ ^ [I4: B,X2: A] : ( member @ A @ X2 @ ( R3 @ I4 ) )
@ S2 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ R3 @ S2 ) ) ) ) ) ).
% INF_INT_eq
thf(fact_6883_SUP__UN__eq,axiom,
! [B: $tType,A: $tType,R3: B > ( set @ A ),S2: set @ B] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image @ B @ ( A > $o )
@ ^ [I4: B,X2: A] : ( member @ A @ X2 @ ( R3 @ I4 ) )
@ S2 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ R3 @ S2 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_6884_Inf__set__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) )
= ( ^ [A4: set @ ( set @ A )] :
( collect @ A
@ ^ [X2: A] : ( complete_Inf_Inf @ $o @ ( image @ ( set @ A ) @ $o @ ( member @ A @ X2 ) @ A4 ) ) ) ) ) ).
% Inf_set_def
thf(fact_6885_Sup__set__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A4: set @ ( set @ A )] :
( collect @ A
@ ^ [X2: A] : ( complete_Sup_Sup @ $o @ ( image @ ( set @ A ) @ $o @ ( member @ A @ X2 ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_6886_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R3: C > ( set @ ( product_prod @ A @ B ) ),S2: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ C @ ( A > B > $o )
@ ^ [I4: C,X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( R3 @ I4 ) )
@ S2 ) )
= ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_6887_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R3: C > ( set @ ( product_prod @ A @ B ) ),S2: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image @ C @ ( A > B > $o )
@ ^ [I4: C,X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( R3 @ I4 ) )
@ S2 ) )
= ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_6888_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S2: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I4: set @ ( product_prod @ A @ B ),X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ I4 )
@ S2 ) )
= ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S2 ) ) ) ) ).
% INF_Int_eq2
thf(fact_6889_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S2: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I4: set @ ( product_prod @ A @ B ),X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ I4 )
@ S2 ) )
= ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S2 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_6890_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X8: set @ nat] :
( if @ nat
@ ( X8
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X8 ) ) ) ) ).
% Sup_nat_def
thf(fact_6891_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A3: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X2: A] : X2
@ A3 ) )
= ( Sup @ A3 ) ) ).
% Sup.SUP_identity_eq
thf(fact_6892_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A3: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X2: A] : X2
@ A3 ) )
= ( Inf @ A3 ) ) ).
% Inf.INF_identity_eq
thf(fact_6893_Sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A4: set @ ( A > B ),X2: A] :
( complete_Sup_Sup @ B
@ ( image @ ( A > B ) @ B
@ ^ [F3: A > B] : ( F3 @ X2 )
@ A4 ) ) ) ) ) ).
% Sup_fun_def
thf(fact_6894_Inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A4: set @ ( A > B ),X2: A] :
( complete_Inf_Inf @ B
@ ( image @ ( A > B ) @ B
@ ^ [F3: A > B] : ( F3 @ X2 )
@ A4 ) ) ) ) ) ).
% Inf_fun_def
thf(fact_6895_Union__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A4: set @ ( set @ A )] :
( collect @ A
@ ^ [X2: A] :
? [Y4: set @ A] :
( ( member @ ( set @ A ) @ Y4 @ A4 )
& ( member @ A @ X2 @ Y4 ) ) ) ) ) ).
% Union_eq
thf(fact_6896_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U2: A,A3: set @ A,V2: A] :
( ( member @ A @ U2 @ A3 )
=> ( ( ord_less_eq @ A @ V2 @ U2 )
=> ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Sup_upper2
thf(fact_6897_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ B2 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_6898_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Sup_upper
thf(fact_6899_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,Z3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ Z3 ) ) ) ).
% Sup_least
thf(fact_6900_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ? [X6: A] :
( ( member @ A @ X6 @ B3 )
& ( ord_less_eq @ A @ A6 @ X6 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ).
% Sup_mono
thf(fact_6901_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,X: A] :
( ! [Y3: A] :
( ( member @ A @ Y3 @ A3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [Y3: A] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ A3 )
=> ( ord_less_eq @ A @ Z4 @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ A3 )
= X ) ) ) ) ).
% Sup_eqI
thf(fact_6902_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,S2: set @ A] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S2 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S2 )
& ( ord_less @ A @ A2 @ X2 ) ) ) ) ) ).
% less_Sup_iff
thf(fact_6903_Inf__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,Z3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ).
% Inf_greatest
thf(fact_6904_le__Inf__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: A,A3: set @ A] :
( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ X2 ) ) ) ) ) ).
% le_Inf_iff
thf(fact_6905_Inf__lower2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U2: A,A3: set @ A,V2: A] :
( ( member @ A @ U2 @ A3 )
=> ( ( ord_less_eq @ A @ U2 @ V2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ V2 ) ) ) ) ).
% Inf_lower2
thf(fact_6906_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ X ) ) ) ).
% Inf_lower
thf(fact_6907_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: set @ A,A3: set @ A] :
( ! [B6: A] :
( ( member @ A @ B6 @ B3 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A3 )
& ( ord_less_eq @ A @ X6 @ B6 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ).
% Inf_mono
thf(fact_6908_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,X: A] :
( ! [I2: A] :
( ( member @ A @ I2 @ A3 )
=> ( ord_less_eq @ A @ X @ I2 ) )
=> ( ! [Y3: A] :
( ! [I3: A] :
( ( member @ A @ I3 @ A3 )
=> ( ord_less_eq @ A @ Y3 @ I3 ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ A3 )
= X ) ) ) ) ).
% Inf_eqI
thf(fact_6909_Inf__less__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [S2: set @ A,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S2 ) @ A2 )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S2 )
& ( ord_less @ A @ X2 @ A2 ) ) ) ) ) ).
% Inf_less_iff
thf(fact_6910_Union__least,axiom,
! [A: $tType,A3: set @ ( set @ A ),C3: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ X10 @ C3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ C3 ) ) ).
% Union_least
thf(fact_6911_Union__upper,axiom,
! [A: $tType,B3: set @ A,A3: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) ) ).
% Union_upper
thf(fact_6912_Union__subsetI,axiom,
! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A3 )
=> ? [Y2: set @ A] :
( ( member @ ( set @ A ) @ Y2 @ B3 )
& ( ord_less_eq @ ( set @ A ) @ X3 @ Y2 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ).
% Union_subsetI
thf(fact_6913_Inter__lower,axiom,
! [A: $tType,B3: set @ A,A3: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B3 ) ) ).
% Inter_lower
thf(fact_6914_Inter__greatest,axiom,
! [A: $tType,A3: set @ ( set @ A ),C3: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ C3 @ X10 ) )
=> ( ord_less_eq @ ( set @ A ) @ C3 @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) ) ) ).
% Inter_greatest
thf(fact_6915_SUP__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > C > A,B3: set @ C,A3: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I4: B] : ( complete_Sup_Sup @ A @ ( image @ C @ A @ ( F2 @ I4 ) @ B3 ) )
@ A3 ) )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [J3: C] :
( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I4: B] : ( F2 @ I4 @ J3 )
@ A3 ) )
@ B3 ) ) ) ) ).
% SUP_commute
thf(fact_6916_INF__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > C > A,B3: set @ C,A3: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I4: B] : ( complete_Inf_Inf @ A @ ( image @ C @ A @ ( F2 @ I4 ) @ B3 ) )
@ A3 ) )
= ( complete_Inf_Inf @ A
@ ( image @ C @ A
@ ^ [J3: C] :
( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I4: B] : ( F2 @ I4 @ J3 )
@ A3 ) )
@ B3 ) ) ) ) ).
% INF_commute
thf(fact_6917_UN__UN__flatten,axiom,
! [A: $tType,B: $tType,C: $tType,C3: B > ( set @ A ),B3: C > ( set @ B ),A3: set @ C] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B3 @ A3 ) ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [Y4: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C3 @ ( B3 @ Y4 ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_6918_UN__E,axiom,
! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A3: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) )
=> ~ ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ~ ( member @ A @ B2 @ ( B3 @ X3 ) ) ) ) ).
% UN_E
thf(fact_6919_UN__extend__simps_I9_J,axiom,
! [S9: $tType,R7: $tType,Q7: $tType,C3: R7 > ( set @ S9 ),B3: Q7 > ( set @ R7 ),A3: set @ Q7] :
( ( complete_Sup_Sup @ ( set @ S9 )
@ ( image @ Q7 @ ( set @ S9 )
@ ^ [X2: Q7] : ( complete_Sup_Sup @ ( set @ S9 ) @ ( image @ R7 @ ( set @ S9 ) @ C3 @ ( B3 @ X2 ) ) )
@ A3 ) )
= ( complete_Sup_Sup @ ( set @ S9 ) @ ( image @ R7 @ ( set @ S9 ) @ C3 @ ( complete_Sup_Sup @ ( set @ R7 ) @ ( image @ Q7 @ ( set @ R7 ) @ B3 @ A3 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_6920_INT__E,axiom,
! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A3: set @ B,A2: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) )
=> ( ~ ( member @ A @ B2 @ ( B3 @ A2 ) )
=> ~ ( member @ B @ A2 @ A3 ) ) ) ).
% INT_E
thf(fact_6921_INT__D,axiom,
! [A: $tType,B: $tType,B2: A,B3: B > ( set @ A ),A3: set @ B,A2: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) )
=> ( ( member @ B @ A2 @ A3 )
=> ( member @ A @ B2 @ ( B3 @ A2 ) ) ) ) ).
% INT_D
thf(fact_6922_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,A3: set @ A] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A3 ) )
= ( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less @ A @ Y4 @ X2 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_6923_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ X )
= ( ! [Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less @ A @ X2 @ Y4 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_6924_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B3: set @ C,F2: B > A,G: C > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ? [X6: C] :
( ( member @ C @ X6 @ B3 )
& ( ord_less_eq @ A @ ( F2 @ I2 ) @ ( G @ X6 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B3 )
=> ? [X6: B] :
( ( member @ B @ X6 @ A3 )
& ( ord_less_eq @ A @ ( G @ J2 ) @ ( F2 @ X6 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_6925_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,U2: A] :
( ! [V4: A] :
( ( member @ A @ V4 @ A3 )
=> ( ord_less_eq @ A @ U2 @ V4 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_6926_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B3: set @ C,G: C > A,F2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ? [X6: C] :
( ( member @ C @ X6 @ B3 )
& ( ord_less_eq @ A @ ( G @ X6 ) @ ( F2 @ I2 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B3 )
=> ? [X6: B] :
( ( member @ B @ X6 @ A3 )
& ( ord_less_eq @ A @ ( F2 @ X6 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ) ).
% INF_eq
thf(fact_6927_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ).
% Sup_subset_mono
thf(fact_6928_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,U2: A] :
( ! [V4: A] :
( ( member @ A @ V4 @ A3 )
=> ( ord_less_eq @ A @ V4 @ U2 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ U2 ) ) ) ) ).
% Inf_less_eq
thf(fact_6929_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ).
% Inf_superset_mono
thf(fact_6930_Union__mono,axiom,
! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ).
% Union_mono
thf(fact_6931_Inter__anti__mono,axiom,
! [A: $tType,B3: set @ ( set @ A ),A3: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B3 ) ) ) ).
% Inter_anti_mono
thf(fact_6932_Inter__subset,axiom,
! [A: $tType,A3: set @ ( set @ A ),B3: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ X10 @ B3 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B3 ) ) ) ).
% Inter_subset
thf(fact_6933_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A,X: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ X ) )
=> ( ! [Y3: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) )
= X ) ) ) ) ).
% SUP_eqI
thf(fact_6934_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B3: set @ C,F2: B > A,G: C > A] :
( ! [N2: B] :
( ( member @ B @ N2 @ A3 )
=> ? [X6: C] :
( ( member @ C @ X6 @ B3 )
& ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ).
% SUP_mono
thf(fact_6935_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A,U2: A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ U2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U2 ) ) ) ).
% SUP_least
thf(fact_6936_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G: B > A,A3: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ).
% SUP_mono'
thf(fact_6937_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A3: set @ B,F2: B > A] :
( ( member @ B @ I @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% SUP_upper
thf(fact_6938_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A3: set @ B,U2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ U2 ) ) ) ) ) ).
% SUP_le_iff
thf(fact_6939_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A3: set @ B,U2: A,F2: B > A] :
( ( member @ B @ I @ A3 )
=> ( ( ord_less_eq @ A @ U2 @ ( F2 @ I ) )
=> ( ord_less_eq @ A @ U2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_6940_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A3: set @ B,Y: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ Y )
=> ( ( member @ B @ I @ A3 )
=> ( ord_less @ A @ ( F2 @ I ) @ Y ) ) ) ) ).
% SUP_lessD
thf(fact_6941_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,F2: B > A,A3: set @ B] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less @ A @ A2 @ ( F2 @ X2 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_6942_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,X: A,F2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ X @ ( F2 @ I2 ) ) )
=> ( ! [Y3: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ Y3 @ ( F2 @ I3 ) ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) )
= X ) ) ) ) ).
% INF_eqI
thf(fact_6943_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: set @ B,A3: set @ C,F2: C > A,G: B > A] :
( ! [M4: B] :
( ( member @ B @ M4 @ B3 )
=> ? [X6: C] :
( ( member @ C @ X6 @ A3 )
& ( ord_less_eq @ A @ ( F2 @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ).
% INF_mono
thf(fact_6944_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A3: set @ B,F2: B > A] :
( ( member @ B @ I @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( F2 @ I ) ) ) ) ).
% INF_lower
thf(fact_6945_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G: B > A,A3: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ).
% INF_mono'
thf(fact_6946_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A3: set @ B,F2: B > A,U2: A] :
( ( member @ B @ I @ A3 )
=> ( ( ord_less_eq @ A @ ( F2 @ I ) @ U2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U2 ) ) ) ) ).
% INF_lower2
thf(fact_6947_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U2: A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ A @ U2 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ord_less_eq @ A @ U2 @ ( F2 @ X2 ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_6948_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,U2: A,F2: B > A] :
( ! [I2: B] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ U2 @ ( F2 @ I2 ) ) )
=> ( ord_less_eq @ A @ U2 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% INF_greatest
thf(fact_6949_subset__Pow__Union,axiom,
! [A: $tType,A3: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A3 @ ( pow2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) ) ).
% subset_Pow_Union
thf(fact_6950_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A,F2: B > A,A3: set @ B,I: B] :
( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
=> ( ( member @ B @ I @ A3 )
=> ( ord_less @ A @ Y @ ( F2 @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_6951_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ A2 )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less @ A @ ( F2 @ X2 ) @ A2 ) ) ) ) ) ).
% INF_less_iff
thf(fact_6952_complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A3: set @ B,G: B > A] :
( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [A5: B] : ( sup_sup @ A @ ( F2 @ A5 ) @ ( G @ A5 ) )
@ A3 ) ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_6953_SUP__absorb,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [K: B,I6: set @ B,A3: B > A] :
( ( member @ B @ K @ I6 )
=> ( ( sup_sup @ A @ ( A3 @ K ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ A3 @ I6 ) ) )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ A3 @ I6 ) ) ) ) ) ).
% SUP_absorb
thf(fact_6954_INF__absorb,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [K: B,I6: set @ B,A3: B > A] :
( ( member @ B @ K @ I6 )
=> ( ( inf_inf @ A @ ( A3 @ K ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ A3 @ I6 ) ) )
= ( complete_Inf_Inf @ A @ ( image @ B @ A @ A3 @ I6 ) ) ) ) ) ).
% INF_absorb
thf(fact_6955_INF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A3: set @ B,G: B > A] :
( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A3 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [A5: B] : ( inf_inf @ A @ ( F2 @ A5 ) @ ( G @ A5 ) )
@ A3 ) ) ) ) ).
% INF_inf_distrib
thf(fact_6956_image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,B3: C > ( set @ B ),A3: set @ C] :
( ( image @ B @ A @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B3 @ A3 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [X2: C] : ( image @ B @ A @ F2 @ ( B3 @ X2 ) )
@ A3 ) ) ) ).
% image_UN
thf(fact_6957_UN__extend__simps_I10_J,axiom,
! [V5: $tType,U5: $tType,T: $tType,B3: U5 > ( set @ V5 ),F2: T > U5,A3: set @ T] :
( ( complete_Sup_Sup @ ( set @ V5 )
@ ( image @ T @ ( set @ V5 )
@ ^ [A5: T] : ( B3 @ ( F2 @ A5 ) )
@ A3 ) )
= ( complete_Sup_Sup @ ( set @ V5 ) @ ( image @ U5 @ ( set @ V5 ) @ B3 @ ( image @ T @ U5 @ F2 @ A3 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_6958_UN__empty2,axiom,
! [B: $tType,A: $tType,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X2: B] : ( bot_bot @ ( set @ A ) )
@ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_6959_UN__empty,axiom,
! [B: $tType,A: $tType,B3: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_6960_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B3 @ X2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_6961_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B3 @ X2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_6962_UN__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ B3 ) ) ) ) ) ).
% UN_mono
thf(fact_6963_UN__least,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: A > ( set @ B ),C3: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( B3 @ X3 ) @ C3 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) @ C3 ) ) ).
% UN_least
thf(fact_6964_UN__upper,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( B3 @ A2 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ).
% UN_upper
thf(fact_6965_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),I6: set @ B,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) @ B3 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I6 )
=> ( ord_less_eq @ ( set @ A ) @ ( A3 @ X2 ) @ B3 ) ) ) ) ).
% UN_subset_iff
thf(fact_6966_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U2: A,A3: set @ A,A2: B,B3: A > ( set @ B )] :
( ( member @ A @ U2 @ A3 )
=> ( ( complete_Sup_Sup @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X2: A] : ( insert @ B @ A2 @ ( B3 @ X2 ) )
@ A3 ) )
= ( insert @ B @ A2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_6967_UN__extend__simps_I5_J,axiom,
! [I5: $tType,J4: $tType,A3: set @ I5,B3: J4 > ( set @ I5 ),C3: set @ J4] :
( ( inf_inf @ ( set @ I5 ) @ A3 @ ( complete_Sup_Sup @ ( set @ I5 ) @ ( image @ J4 @ ( set @ I5 ) @ B3 @ C3 ) ) )
= ( complete_Sup_Sup @ ( set @ I5 )
@ ( image @ J4 @ ( set @ I5 )
@ ^ [X2: J4] : ( inf_inf @ ( set @ I5 ) @ A3 @ ( B3 @ X2 ) )
@ C3 ) ) ) ).
% UN_extend_simps(5)
thf(fact_6968_UN__extend__simps_I4_J,axiom,
! [H5: $tType,G5: $tType,A3: G5 > ( set @ H5 ),C3: set @ G5,B3: set @ H5] :
( ( inf_inf @ ( set @ H5 ) @ ( complete_Sup_Sup @ ( set @ H5 ) @ ( image @ G5 @ ( set @ H5 ) @ A3 @ C3 ) ) @ B3 )
= ( complete_Sup_Sup @ ( set @ H5 )
@ ( image @ G5 @ ( set @ H5 )
@ ^ [X2: G5] : ( inf_inf @ ( set @ H5 ) @ ( A3 @ X2 ) @ B3 )
@ C3 ) ) ) ).
% UN_extend_simps(4)
thf(fact_6969_Int__UN__distrib,axiom,
! [A: $tType,B: $tType,B3: set @ A,A3: B > ( set @ A ),I6: set @ B] :
( ( inf_inf @ ( set @ A ) @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I4: B] : ( inf_inf @ ( set @ A ) @ B3 @ ( A3 @ I4 ) )
@ I6 ) ) ) ).
% Int_UN_distrib
thf(fact_6970_Int__UN__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A3: B > ( set @ A ),I6: set @ B,B3: C > ( set @ A ),J5: set @ C] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ C @ ( set @ A ) @ B3 @ J5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I4: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [J3: C] : ( inf_inf @ ( set @ A ) @ ( A3 @ I4 ) @ ( B3 @ J3 ) )
@ J5 ) )
@ I6 ) ) ) ).
% Int_UN_distrib2
thf(fact_6971_INT__extend__simps_I10_J,axiom,
! [V5: $tType,U5: $tType,T: $tType,B3: U5 > ( set @ V5 ),F2: T > U5,A3: set @ T] :
( ( complete_Inf_Inf @ ( set @ V5 )
@ ( image @ T @ ( set @ V5 )
@ ^ [A5: T] : ( B3 @ ( F2 @ A5 ) )
@ A3 ) )
= ( complete_Inf_Inf @ ( set @ V5 ) @ ( image @ U5 @ ( set @ V5 ) @ B3 @ ( image @ T @ U5 @ F2 @ A3 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_6972_Un__Union__image,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),B3: B > ( set @ A ),C3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X2: B] : ( sup_sup @ ( set @ A ) @ ( A3 @ X2 ) @ ( B3 @ X2 ) )
@ C3 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ C3 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C3 ) ) ) ) ).
% Un_Union_image
thf(fact_6973_UN__Un__distrib,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),B3: B > ( set @ A ),I6: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I4: B] : ( sup_sup @ ( set @ A ) @ ( A3 @ I4 ) @ ( B3 @ I4 ) )
@ I6 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ I6 ) ) ) ) ).
% UN_Un_distrib
thf(fact_6974_UN__absorb,axiom,
! [B: $tType,A: $tType,K: A,I6: set @ A,A3: A > ( set @ B )] :
( ( member @ A @ K @ I6 )
=> ( ( sup_sup @ ( set @ B ) @ ( A3 @ K ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) ) ).
% UN_absorb
thf(fact_6975_UN__extend__simps_I6_J,axiom,
! [L5: $tType,K8: $tType,A3: K8 > ( set @ L5 ),C3: set @ K8,B3: set @ L5] :
( ( minus_minus @ ( set @ L5 ) @ ( complete_Sup_Sup @ ( set @ L5 ) @ ( image @ K8 @ ( set @ L5 ) @ A3 @ C3 ) ) @ B3 )
= ( complete_Sup_Sup @ ( set @ L5 )
@ ( image @ K8 @ ( set @ L5 )
@ ^ [X2: K8] : ( minus_minus @ ( set @ L5 ) @ ( A3 @ X2 ) @ B3 )
@ C3 ) ) ) ).
% UN_extend_simps(6)
thf(fact_6976_INT__lower,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B3: A > ( set @ B )] :
( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) @ ( B3 @ A2 ) ) ) ).
% INT_lower
thf(fact_6977_INT__greatest,axiom,
! [B: $tType,A: $tType,A3: set @ A,C3: set @ B,B3: A > ( set @ B )] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ C3 @ ( B3 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ C3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ).
% INT_greatest
thf(fact_6978_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ B3 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ A3 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_6979_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B3: set @ A,A3: B > ( set @ A ),I6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I6 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ ( A3 @ X2 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_6980_INT__extend__simps_I5_J,axiom,
! [I5: $tType,J4: $tType,A2: I5,B3: J4 > ( set @ I5 ),C3: set @ J4] :
( ( insert @ I5 @ A2 @ ( complete_Inf_Inf @ ( set @ I5 ) @ ( image @ J4 @ ( set @ I5 ) @ B3 @ C3 ) ) )
= ( complete_Inf_Inf @ ( set @ I5 )
@ ( image @ J4 @ ( set @ I5 )
@ ^ [X2: J4] : ( insert @ I5 @ A2 @ ( B3 @ X2 ) )
@ C3 ) ) ) ).
% INT_extend_simps(5)
thf(fact_6981_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U2: A,A3: set @ A,A2: B,B3: A > ( set @ B )] :
( ( member @ A @ U2 @ A3 )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X2: A] : ( insert @ B @ A2 @ ( B3 @ X2 ) )
@ A3 ) )
= ( insert @ B @ A2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_6982_Collect__bex__eq,axiom,
! [A: $tType,B: $tType,A3: set @ B,P: A > B > $o] :
( ( collect @ A
@ ^ [X2: A] :
? [Y4: B] :
( ( member @ B @ Y4 @ A3 )
& ( P @ X2 @ Y4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y4: B] :
( collect @ A
@ ^ [X2: A] : ( P @ X2 @ Y4 ) )
@ A3 ) ) ) ).
% Collect_bex_eq
thf(fact_6983_UNION__eq,axiom,
! [B: $tType,A: $tType,B3: B > ( set @ A ),A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) )
= ( collect @ A
@ ^ [Y4: A] :
? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( member @ A @ Y4 @ ( B3 @ X2 ) ) ) ) ) ).
% UNION_eq
thf(fact_6984_INT__absorb,axiom,
! [B: $tType,A: $tType,K: A,I6: set @ A,A3: A > ( set @ B )] :
( ( member @ A @ K @ I6 )
=> ( ( inf_inf @ ( set @ B ) @ ( A3 @ K ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) )
= ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) ) ).
% INT_absorb
thf(fact_6985_INT__Int__distrib,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),B3: B > ( set @ A ),I6: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I4: B] : ( inf_inf @ ( set @ A ) @ ( A3 @ I4 ) @ ( B3 @ I4 ) )
@ I6 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ I6 ) ) ) ) ).
% INT_Int_distrib
thf(fact_6986_Int__Inter__image,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),B3: B > ( set @ A ),C3: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X2: B] : ( inf_inf @ ( set @ A ) @ ( A3 @ X2 ) @ ( B3 @ X2 ) )
@ C3 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ C3 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C3 ) ) ) ) ).
% Int_Inter_image
thf(fact_6987_Un__INT__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A3: B > ( set @ A ),I6: set @ B,B3: C > ( set @ A ),J5: set @ C] :
( ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ C @ ( set @ A ) @ B3 @ J5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I4: B] :
( complete_Inf_Inf @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [J3: C] : ( sup_sup @ ( set @ A ) @ ( A3 @ I4 ) @ ( B3 @ J3 ) )
@ J5 ) )
@ I6 ) ) ) ).
% Un_INT_distrib2
thf(fact_6988_Un__INT__distrib,axiom,
! [A: $tType,B: $tType,B3: set @ A,A3: B > ( set @ A ),I6: set @ B] :
( ( sup_sup @ ( set @ A ) @ B3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I4: B] : ( sup_sup @ ( set @ A ) @ B3 @ ( A3 @ I4 ) )
@ I6 ) ) ) ).
% Un_INT_distrib
thf(fact_6989_INT__extend__simps_I6_J,axiom,
! [L5: $tType,K8: $tType,A3: K8 > ( set @ L5 ),C3: set @ K8,B3: set @ L5] :
( ( sup_sup @ ( set @ L5 ) @ ( complete_Inf_Inf @ ( set @ L5 ) @ ( image @ K8 @ ( set @ L5 ) @ A3 @ C3 ) ) @ B3 )
= ( complete_Inf_Inf @ ( set @ L5 )
@ ( image @ K8 @ ( set @ L5 )
@ ^ [X2: K8] : ( sup_sup @ ( set @ L5 ) @ ( A3 @ X2 ) @ B3 )
@ C3 ) ) ) ).
% INT_extend_simps(6)
thf(fact_6990_INT__extend__simps_I7_J,axiom,
! [M11: $tType,N11: $tType,A3: set @ M11,B3: N11 > ( set @ M11 ),C3: set @ N11] :
( ( sup_sup @ ( set @ M11 ) @ A3 @ ( complete_Inf_Inf @ ( set @ M11 ) @ ( image @ N11 @ ( set @ M11 ) @ B3 @ C3 ) ) )
= ( complete_Inf_Inf @ ( set @ M11 )
@ ( image @ N11 @ ( set @ M11 )
@ ^ [X2: N11] : ( sup_sup @ ( set @ M11 ) @ A3 @ ( B3 @ X2 ) )
@ C3 ) ) ) ).
% INT_extend_simps(7)
thf(fact_6991_INT__extend__simps_I9_J,axiom,
! [S9: $tType,R7: $tType,Q7: $tType,C3: R7 > ( set @ S9 ),B3: Q7 > ( set @ R7 ),A3: set @ Q7] :
( ( complete_Inf_Inf @ ( set @ S9 )
@ ( image @ Q7 @ ( set @ S9 )
@ ^ [X2: Q7] : ( complete_Inf_Inf @ ( set @ S9 ) @ ( image @ R7 @ ( set @ S9 ) @ C3 @ ( B3 @ X2 ) ) )
@ A3 ) )
= ( complete_Inf_Inf @ ( set @ S9 ) @ ( image @ R7 @ ( set @ S9 ) @ C3 @ ( complete_Sup_Sup @ ( set @ R7 ) @ ( image @ Q7 @ ( set @ R7 ) @ B3 @ A3 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_6992_image__Union,axiom,
! [A: $tType,B: $tType,F2: B > A,S2: set @ ( set @ B )] :
( ( image @ B @ A @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ S2 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ S2 ) ) ) ).
% image_Union
thf(fact_6993_Pow__INT__eq,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] :
( ( pow2 @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) )
= ( complete_Inf_Inf @ ( set @ ( set @ A ) )
@ ( image @ B @ ( set @ ( set @ A ) )
@ ^ [X2: B] : ( pow2 @ A @ ( B3 @ X2 ) )
@ A3 ) ) ) ).
% Pow_INT_eq
thf(fact_6994_Int__Union,axiom,
! [A: $tType,A3: set @ A,B3: set @ ( set @ A )] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 ) @ B3 ) ) ) ).
% Int_Union
thf(fact_6995_Int__Union2,axiom,
! [A: $tType,B3: set @ ( set @ A ),A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) @ A3 )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ ( set @ A ) @ ( set @ A )
@ ^ [C4: set @ A] : ( inf_inf @ ( set @ A ) @ C4 @ A3 )
@ B3 ) ) ) ).
% Int_Union2
thf(fact_6996_UN__extend__simps_I8_J,axiom,
! [P9: $tType,O2: $tType,B3: O2 > ( set @ P9 ),A3: set @ ( set @ O2 )] :
( ( complete_Sup_Sup @ ( set @ P9 )
@ ( image @ ( set @ O2 ) @ ( set @ P9 )
@ ^ [Y4: set @ O2] : ( complete_Sup_Sup @ ( set @ P9 ) @ ( image @ O2 @ ( set @ P9 ) @ B3 @ Y4 ) )
@ A3 ) )
= ( complete_Sup_Sup @ ( set @ P9 ) @ ( image @ O2 @ ( set @ P9 ) @ B3 @ ( complete_Sup_Sup @ ( set @ O2 ) @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_6997_Un__Inter,axiom,
! [A: $tType,A3: set @ A,B3: set @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B3 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 ) @ B3 ) ) ) ).
% Un_Inter
thf(fact_6998_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,F2: B > A,A3: set @ B] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) )
= ( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X )
=> ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less @ A @ Y4 @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_6999_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ X )
= ( ! [Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less @ A @ ( F2 @ X2 ) @ Y4 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_7000_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,C2: A,F2: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( ord_less_eq @ A @ C2 @ ( F2 @ I2 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ I6 ) )
= C2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I6 )
=> ( ( F2 @ X2 )
= C2 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_7001_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F2: B > A,C2: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( ord_less_eq @ A @ ( F2 @ I2 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) )
= C2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I6 )
=> ( ( F2 @ X2 )
= C2 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_7002_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Inf_le_Sup
thf(fact_7003_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B3: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_7004_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B3: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_7005_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B3: set @ B,F2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_7006_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: set @ B,A3: set @ B,F2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_7007_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y4: B] : C2
@ A3 ) )
= ( bot_bot @ A ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y4: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% SUP_constant
thf(fact_7008_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_7009_uminus__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B3: B > A,A3: set @ B] :
( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ B3 @ A3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( B3 @ X2 ) )
@ A3 ) ) ) ) ).
% uminus_INF
thf(fact_7010_uminus__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B3: B > A,A3: set @ B] :
( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ B3 @ A3 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( B3 @ X2 ) )
@ A3 ) ) ) ) ).
% uminus_SUP
thf(fact_7011_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,X: A,F2: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ X @ ( F2 @ I4 ) )
@ I6 ) )
= ( inf_inf @ A @ X @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_7012_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F2: B > A,X: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ ( F2 @ I4 ) @ X )
@ I6 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) ) @ X ) ) ) ) ).
% INF_inf_const2
thf(fact_7013_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A2: B,A3: set @ B] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( insert @ B @ A2 @ A3 ) ) )
= ( sup_sup @ A @ ( F2 @ A2 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% SUP_insert
thf(fact_7014_INF__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A2: B,A3: set @ B] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( insert @ B @ A2 @ A3 ) ) )
= ( inf_inf @ A @ ( F2 @ A2 ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% INF_insert
thf(fact_7015_SUP__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M7: B > A,A3: set @ B,B3: set @ B] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ M7 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ M7 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ M7 @ B3 ) ) ) ) ) ).
% SUP_union
thf(fact_7016_INF__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M7: B > A,A3: set @ B,B3: set @ B] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ M7 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ M7 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ M7 @ B3 ) ) ) ) ) ).
% INF_union
thf(fact_7017_SUP__UNION,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,G: C > ( set @ B ),A3: set @ C] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ G @ A3 ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [Y4: C] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( G @ Y4 ) ) )
@ A3 ) ) ) ) ).
% SUP_UNION
thf(fact_7018_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C3: set @ B,A2: A,B3: B > ( set @ A )] :
( ( ( C3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C3 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C3 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X2: B] : ( insert @ A @ A2 @ ( B3 @ X2 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_7019_UN__extend__simps_I2_J,axiom,
! [D3: $tType,C: $tType,C3: set @ C,A3: C > ( set @ D3 ),B3: set @ D3] :
( ( ( C3
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D3 ) @ ( complete_Sup_Sup @ ( set @ D3 ) @ ( image @ C @ ( set @ D3 ) @ A3 @ C3 ) ) @ B3 )
= B3 ) )
& ( ( C3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D3 ) @ ( complete_Sup_Sup @ ( set @ D3 ) @ ( image @ C @ ( set @ D3 ) @ A3 @ C3 ) ) @ B3 )
= ( complete_Sup_Sup @ ( set @ D3 )
@ ( image @ C @ ( set @ D3 )
@ ^ [X2: C] : ( sup_sup @ ( set @ D3 ) @ ( A3 @ X2 ) @ B3 )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_7020_UN__extend__simps_I3_J,axiom,
! [E4: $tType,F: $tType,C3: set @ F,A3: set @ E4,B3: F > ( set @ E4 )] :
( ( ( C3
= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E4 ) @ A3 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image @ F @ ( set @ E4 ) @ B3 @ C3 ) ) )
= A3 ) )
& ( ( C3
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E4 ) @ A3 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image @ F @ ( set @ E4 ) @ B3 @ C3 ) ) )
= ( complete_Sup_Sup @ ( set @ E4 )
@ ( image @ F @ ( set @ E4 )
@ ^ [X2: F] : ( sup_sup @ ( set @ E4 ) @ A3 @ ( B3 @ X2 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_7021_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C3: set @ A,A3: A > ( set @ B ),B3: set @ B] :
( ( ( C3
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C3 ) ) @ B3 )
= B3 ) )
& ( ( C3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C3 ) ) @ B3 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X2 ) @ B3 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_7022_INT__extend__simps_I2_J,axiom,
! [C: $tType,D3: $tType,C3: set @ D3,A3: set @ C,B3: D3 > ( set @ C )] :
( ( ( C3
= ( bot_bot @ ( set @ D3 ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D3 @ ( set @ C ) @ B3 @ C3 ) ) )
= A3 ) )
& ( ( C3
!= ( bot_bot @ ( set @ D3 ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D3 @ ( set @ C ) @ B3 @ C3 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image @ D3 @ ( set @ C )
@ ^ [X2: D3] : ( inf_inf @ ( set @ C ) @ A3 @ ( B3 @ X2 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_7023_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B ),F2: B > C,A9: A > ( set @ C )] :
( ! [I2: A,J2: A] :
( ( member @ A @ I2 @ I6 )
=> ( ( member @ A @ J2 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A3 @ I2 ) @ ( A3 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A3 @ J2 ) @ ( A3 @ I2 ) ) ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( bij_betw @ B @ C @ F2 @ ( A3 @ I2 ) @ ( A9 @ I2 ) ) )
=> ( bij_betw @ B @ C @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ A @ ( set @ C ) @ A9 @ I6 ) ) ) ) ) ).
% bij_betw_UNION_chain
thf(fact_7024_INT__Un,axiom,
! [A: $tType,B: $tType,M7: B > ( set @ A ),A3: set @ B,B3: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ M7 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ M7 @ A3 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ M7 @ B3 ) ) ) ) ).
% INT_Un
thf(fact_7025_UN__extend__simps_I7_J,axiom,
! [M11: $tType,N11: $tType,A3: set @ M11,B3: N11 > ( set @ M11 ),C3: set @ N11] :
( ( minus_minus @ ( set @ M11 ) @ A3 @ ( complete_Inf_Inf @ ( set @ M11 ) @ ( image @ N11 @ ( set @ M11 ) @ B3 @ C3 ) ) )
= ( complete_Sup_Sup @ ( set @ M11 )
@ ( image @ N11 @ ( set @ M11 )
@ ^ [X2: N11] : ( minus_minus @ ( set @ M11 ) @ A3 @ ( B3 @ X2 ) )
@ C3 ) ) ) ).
% UN_extend_simps(7)
thf(fact_7026_Union__Int__subset,axiom,
! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A3 @ B3 ) ) @ ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ).
% Union_Int_subset
thf(fact_7027_Inter__Un__subset,axiom,
! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B3 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A3 @ B3 ) ) ) ).
% Inter_Un_subset
thf(fact_7028_INT__extend__simps_I8_J,axiom,
! [P9: $tType,O2: $tType,B3: O2 > ( set @ P9 ),A3: set @ ( set @ O2 )] :
( ( complete_Inf_Inf @ ( set @ P9 )
@ ( image @ ( set @ O2 ) @ ( set @ P9 )
@ ^ [Y4: set @ O2] : ( complete_Inf_Inf @ ( set @ P9 ) @ ( image @ O2 @ ( set @ P9 ) @ B3 @ Y4 ) )
@ A3 ) )
= ( complete_Inf_Inf @ ( set @ P9 ) @ ( image @ O2 @ ( set @ P9 ) @ B3 @ ( complete_Sup_Sup @ ( set @ O2 ) @ A3 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_7029_UN__Pow__subset,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] :
( ord_less_eq @ ( set @ ( set @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image @ B @ ( set @ ( set @ A ) )
@ ^ [X2: B] : ( pow2 @ A @ ( B3 @ X2 ) )
@ A3 ) )
@ ( pow2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% UN_Pow_subset
thf(fact_7030_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B14: set @ ( set @ A ),A3: set @ A] :
( ( ( B14
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B14 ) )
= A3 ) )
& ( ( B14
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B14 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 ) @ B14 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_7031_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B14: set @ ( set @ A ),A3: set @ A] :
( ( ( B14
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B14 ) @ A3 )
= A3 ) )
& ( ( B14
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B14 ) @ A3 )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ ( set @ A ) @ ( set @ A )
@ ^ [B4: set @ A] : ( inf_inf @ ( set @ A ) @ B4 @ A3 )
@ B14 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_7032_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F2: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_7033_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X2: B] : ( insert @ A @ ( F2 @ X2 ) @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) )
= ( image @ B @ A @ F2 @ A3 ) ) ).
% UNION_singleton_eq_range
thf(fact_7034_INT__extend__simps_I4_J,axiom,
! [G5: $tType,H5: $tType,C3: set @ H5,A3: set @ G5,B3: H5 > ( set @ G5 )] :
( ( ( C3
= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( minus_minus @ ( set @ G5 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G5 ) @ ( image @ H5 @ ( set @ G5 ) @ B3 @ C3 ) ) )
= A3 ) )
& ( ( C3
!= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( minus_minus @ ( set @ G5 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G5 ) @ ( image @ H5 @ ( set @ G5 ) @ B3 @ C3 ) ) )
= ( complete_Inf_Inf @ ( set @ G5 )
@ ( image @ H5 @ ( set @ G5 )
@ ^ [X2: H5] : ( minus_minus @ ( set @ G5 ) @ A3 @ ( B3 @ X2 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_7035_ccSUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F2: B > A] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_empty
thf(fact_7036_ccINF__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: set @ B,F2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I4: B] : F2
@ A3 ) )
= F2 ) ) ) ).
% ccINF_const
thf(fact_7037_SUP2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A2: A,A3: set @ A,B3: A > B > C > $o,B2: B,C2: C] :
( ( member @ A @ A2 @ A3 )
=> ( ( B3 @ A2 @ B2 @ C2 )
=> ( complete_Sup_Sup @ ( B > C > $o ) @ ( image @ A @ ( B > C > $o ) @ B3 @ A3 ) @ B2 @ C2 ) ) ) ).
% SUP2_I
thf(fact_7038_INF1__I,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: A > B > $o,B2: B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( B3 @ X3 @ B2 ) )
=> ( complete_Inf_Inf @ ( B > $o ) @ ( image @ A @ ( B > $o ) @ B3 @ A3 ) @ B2 ) ) ).
% INF1_I
thf(fact_7039_INF2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A3: set @ A,B3: A > B > C > $o,B2: B,C2: C] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( B3 @ X3 @ B2 @ C2 ) )
=> ( complete_Inf_Inf @ ( B > C > $o ) @ ( image @ A @ ( B > C > $o ) @ B3 @ A3 ) @ B2 @ C2 ) ) ).
% INF2_I
thf(fact_7040_SUP1__I,axiom,
! [A: $tType,B: $tType,A2: A,A3: set @ A,B3: A > B > $o,B2: B] :
( ( member @ A @ A2 @ A3 )
=> ( ( B3 @ A2 @ B2 )
=> ( complete_Sup_Sup @ ( B > $o ) @ ( image @ A @ ( B > $o ) @ B3 @ A3 ) @ B2 ) ) ) ).
% SUP1_I
thf(fact_7041_ccSUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( bot_bot @ A )
@ A3 ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_bot
thf(fact_7042_ccSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: set @ B,F2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I4: B] : F2
@ A3 ) )
= F2 ) ) ) ).
% ccSUP_const
thf(fact_7043_INF1__D,axiom,
! [B: $tType,A: $tType,B3: B > A > $o,A3: set @ B,B2: A,A2: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image @ B @ ( A > $o ) @ B3 @ A3 ) @ B2 )
=> ( ( member @ B @ A2 @ A3 )
=> ( B3 @ A2 @ B2 ) ) ) ).
% INF1_D
thf(fact_7044_INF1__E,axiom,
! [A: $tType,B: $tType,B3: B > A > $o,A3: set @ B,B2: A,A2: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image @ B @ ( A > $o ) @ B3 @ A3 ) @ B2 )
=> ( ~ ( B3 @ A2 @ B2 )
=> ~ ( member @ B @ A2 @ A3 ) ) ) ).
% INF1_E
thf(fact_7045_INF2__D,axiom,
! [A: $tType,C: $tType,B: $tType,B3: C > A > B > $o,A3: set @ C,B2: A,C2: B,A2: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image @ C @ ( A > B > $o ) @ B3 @ A3 ) @ B2 @ C2 )
=> ( ( member @ C @ A2 @ A3 )
=> ( B3 @ A2 @ B2 @ C2 ) ) ) ).
% INF2_D
thf(fact_7046_INF2__E,axiom,
! [B: $tType,A: $tType,C: $tType,B3: C > A > B > $o,A3: set @ C,B2: A,C2: B,A2: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image @ C @ ( A > B > $o ) @ B3 @ A3 ) @ B2 @ C2 )
=> ( ~ ( B3 @ A2 @ B2 @ C2 )
=> ~ ( member @ C @ A2 @ A3 ) ) ) ).
% INF2_E
thf(fact_7047_SUP1__E,axiom,
! [B: $tType,A: $tType,B3: B > A > $o,A3: set @ B,B2: A] :
( ( complete_Sup_Sup @ ( A > $o ) @ ( image @ B @ ( A > $o ) @ B3 @ A3 ) @ B2 )
=> ~ ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ~ ( B3 @ X3 @ B2 ) ) ) ).
% SUP1_E
thf(fact_7048_SUP2__E,axiom,
! [A: $tType,C: $tType,B: $tType,B3: C > A > B > $o,A3: set @ C,B2: A,C2: B] :
( ( complete_Sup_Sup @ ( A > B > $o ) @ ( image @ C @ ( A > B > $o ) @ B3 @ A3 ) @ B2 @ C2 )
=> ~ ! [X3: C] :
( ( member @ C @ X3 @ A3 )
=> ~ ( B3 @ X3 @ B2 @ C2 ) ) ) ).
% SUP2_E
thf(fact_7049_ccpo__Sup__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% ccpo_Sup_singleton
thf(fact_7050_INF__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: A,B3: A] :
( ( inf_inf @ A @ A3
@ ( complete_Inf_Inf @ A
@ ( image @ nat @ A
@ ^ [X2: nat] : B3
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( inf_inf @ A @ A3 @ B3 ) ) ) ).
% INF_nat_binary
thf(fact_7051_SUP__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: A,B3: A] :
( ( sup_sup @ A @ A3
@ ( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [X2: nat] : B3
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% SUP_nat_binary
thf(fact_7052_Inf__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A2: A,A3: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ A3 ) )
= ( inf_inf @ A @ A2 @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ).
% Inf_finite_insert
thf(fact_7053_Sup__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A2: A,A3: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ A3 ) )
= ( sup_sup @ A @ A2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Sup_finite_insert
thf(fact_7054_UN__UN__split__split__eq,axiom,
! [A: $tType,E4: $tType,D3: $tType,C: $tType,B: $tType,A3: B > C > D3 > E4 > ( set @ A ),Y6: set @ ( product_prod @ D3 @ E4 ),X5: set @ ( product_prod @ B @ C )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ ( product_prod @ B @ C ) @ ( set @ A )
@ ( product_case_prod @ B @ C @ ( set @ A )
@ ^ [X12: B,X24: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( product_prod @ D3 @ E4 ) @ ( set @ A ) @ ( product_case_prod @ D3 @ E4 @ ( set @ A ) @ ( A3 @ X12 @ X24 ) ) @ Y6 ) ) )
@ X5 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ ( product_prod @ B @ C ) @ ( set @ A )
@ ^ [X2: product_prod @ B @ C] :
( complete_Sup_Sup @ ( set @ A )
@ ( image @ ( product_prod @ D3 @ E4 ) @ ( set @ A )
@ ^ [Y4: product_prod @ D3 @ E4] :
( product_case_prod @ B @ C @ ( set @ A )
@ ^ [X12: B,X24: C] : ( product_case_prod @ D3 @ E4 @ ( set @ A ) @ ( A3 @ X12 @ X24 ) @ Y4 )
@ X2 )
@ Y6 ) )
@ X5 ) ) ) ).
% UN_UN_split_split_eq
thf(fact_7055_suminf__eq__SUP__real,axiom,
! [X5: nat > real] :
( ( summable @ real @ X5 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( X5 @ I2 ) )
=> ( ( suminf @ real @ X5 )
= ( complete_Sup_Sup @ real
@ ( image @ nat @ real
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ real @ X5 @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% suminf_eq_SUP_real
thf(fact_7056_top__apply,axiom,
! [C: $tType,D3: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D3 > C ) )
= ( ^ [X2: D3] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_7057_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_7058_Int__UNIV,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 )
= ( top_top @ ( set @ A ) ) )
= ( ( A3
= ( top_top @ ( set @ A ) ) )
& ( B3
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_7059_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_7060_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_7061_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_7062_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S8: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S8: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_7063_finite__Collect__not,axiom,
! [A: $tType,P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_7064_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( ( complete_Sup_Sup @ A @ A3 )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A3 )
& ( ord_less @ A @ X2 @ Y4 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_7065_Diff__UNIV,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_7066_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_7067_INF__top__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: B > A,A3: set @ B] :
( ( ( top_top @ A )
= ( complete_Inf_Inf @ A @ ( image @ B @ A @ B3 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B3 @ X2 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(2)
thf(fact_7068_INF__top__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B3: B > A,A3: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ B3 @ A3 ) )
= ( top_top @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B3 @ X2 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(1)
thf(fact_7069_INF__top,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( top_top @ A )
@ A3 ) )
= ( top_top @ A ) ) ) ).
% INF_top
thf(fact_7070_ccINF__top,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( top_top @ A )
@ A3 ) )
= ( top_top @ A ) ) ) ).
% ccINF_top
thf(fact_7071_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A3: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y4: B] :
( ( member @ B @ Y4 @ A3 )
& ( ord_less @ A @ X2 @ ( F2 @ Y4 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_7072_range__constant,axiom,
! [B: $tType,A: $tType,X: A] :
( ( image @ B @ A
@ ^ [Uu: B] : X
@ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_7073_ccINF__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F2: B > A] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% ccINF_empty
thf(fact_7074_INT__constant,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y4: B] : C2
@ A3 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y4: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_7075_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_7076_INT__simps_I2_J,axiom,
! [C: $tType,D3: $tType,C3: set @ D3,A3: set @ C,B3: D3 > ( set @ C )] :
( ( ( C3
= ( bot_bot @ ( set @ D3 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image @ D3 @ ( set @ C )
@ ^ [X2: D3] : ( inf_inf @ ( set @ C ) @ A3 @ ( B3 @ X2 ) )
@ C3 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ D3 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image @ D3 @ ( set @ C )
@ ^ [X2: D3] : ( inf_inf @ ( set @ C ) @ A3 @ ( B3 @ X2 ) )
@ C3 ) )
= ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D3 @ ( set @ C ) @ B3 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_7077_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C3: set @ A,A3: A > ( set @ B ),B3: set @ B] :
( ( ( C3
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X2 ) @ B3 )
@ C3 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X2 ) @ B3 )
@ C3 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C3 ) ) @ B3 ) ) ) ) ).
% INT_simps(1)
thf(fact_7078_INT__simps_I3_J,axiom,
! [E4: $tType,F: $tType,C3: set @ E4,A3: E4 > ( set @ F ),B3: set @ F] :
( ( ( C3
= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E4 @ ( set @ F )
@ ^ [X2: E4] : ( minus_minus @ ( set @ F ) @ ( A3 @ X2 ) @ B3 )
@ C3 ) )
= ( top_top @ ( set @ F ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E4 @ ( set @ F )
@ ^ [X2: E4] : ( minus_minus @ ( set @ F ) @ ( A3 @ X2 ) @ B3 )
@ C3 ) )
= ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E4 @ ( set @ F ) @ A3 @ C3 ) ) @ B3 ) ) ) ) ).
% INT_simps(3)
thf(fact_7079_INT__simps_I4_J,axiom,
! [G5: $tType,H5: $tType,C3: set @ H5,A3: set @ G5,B3: H5 > ( set @ G5 )] :
( ( ( C3
= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G5 )
@ ( image @ H5 @ ( set @ G5 )
@ ^ [X2: H5] : ( minus_minus @ ( set @ G5 ) @ A3 @ ( B3 @ X2 ) )
@ C3 ) )
= ( top_top @ ( set @ G5 ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G5 )
@ ( image @ H5 @ ( set @ G5 )
@ ^ [X2: H5] : ( minus_minus @ ( set @ G5 ) @ A3 @ ( B3 @ X2 ) )
@ C3 ) )
= ( minus_minus @ ( set @ G5 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G5 ) @ ( image @ H5 @ ( set @ G5 ) @ B3 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_7080_sums__SUP,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( canoni5634975068530333245id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( sums @ A @ F2
@ ( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% sums_SUP
thf(fact_7081_UN__lessThan__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_lessThan_UNIV
thf(fact_7082_UN__atMost__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atMost_UNIV
thf(fact_7083_SUP__INF,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P: C > B > A] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y4: B] :
( complete_Inf_Inf @ A
@ ( image @ C @ A
@ ^ [X2: C] : ( P @ X2 @ Y4 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ ( B > C ) @ A
@ ^ [X2: B > C] :
( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y4: B] : ( P @ ( X2 @ Y4 ) @ Y4 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% SUP_INF
thf(fact_7084_INF__SUP,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P: C > B > A] :
( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y4: B] :
( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [X2: C] : ( P @ X2 @ Y4 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ ( B > C ) @ A
@ ^ [F3: B > C] :
( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( P @ ( F3 @ X2 ) @ X2 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% INF_SUP
thf(fact_7085_range__subsetD,axiom,
! [B: $tType,A: $tType,F2: B > A,B3: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) @ B3 )
=> ( member @ A @ ( F2 @ I ) @ B3 ) ) ).
% range_subsetD
thf(fact_7086_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,X: B] :
( ( B2
= ( F2 @ X ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_7087_rangeI,axiom,
! [A: $tType,B: $tType,F2: B > A,X: B] : ( member @ A @ ( F2 @ X ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_7088_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A] :
( ( member @ A @ B2 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X3: B] :
( B2
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_7089_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F2: C > A,G: B > C] :
( ( image @ B @ A
@ ^ [X2: B] : ( F2 @ ( G @ X2 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F2 @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_7090_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G: B > A,F2: A > C] :
( ( finite_finite2 @ A @ ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image @ B @ C
@ ^ [X2: B] : ( F2 @ ( G @ X2 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_7091_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > A,X5: set @ B,G1: A > C,G22: A > C] :
( ( ( image @ B @ A @ F2 @ X5 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ X5 )
=> ( ( comp @ A @ C @ B @ G1 @ F2 @ X3 )
= ( comp @ A @ C @ B @ G22 @ F2 @ X3 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_7092_INTER__UNIV__conv_I1_J,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] :
( ( ( top_top @ ( set @ A ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B3 @ X2 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_7093_INTER__UNIV__conv_I2_J,axiom,
! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] :
( ( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( ( B3 @ X2 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_7094_not__UNIV__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H2 ) ) ) ).
% not_UNIV_le_Iic
thf(fact_7095_not__UNIV__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) ) ) ).
% not_UNIV_le_Icc
thf(fact_7096_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
=> ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_7097_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
= ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_7098_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_7099_subset__UNIV,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_7100_Un__UNIV__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_7101_Un__UNIV__left,axiom,
! [A: $tType,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_7102_Int__UNIV__right,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= A3 ) ).
% Int_UNIV_right
thf(fact_7103_Int__UNIV__left,axiom,
! [A: $tType,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
= B3 ) ).
% Int_UNIV_left
thf(fact_7104_UNIV__eq__I,axiom,
! [A: $tType,A3: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A3 )
=> ( ( top_top @ ( set @ A ) )
= A3 ) ) ).
% UNIV_eq_I
thf(fact_7105_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_7106_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $true ) ) ).
% UNIV_def
thf(fact_7107_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_7108_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_7109_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( A2
!= ( top_top @ A ) )
= ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_7110_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% top.extremum_strict
thf(fact_7111_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_7112_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_7113_Compl__partition2,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ A3 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_7114_Compl__partition,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_7115_Compl__eq__Diff__UNIV,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_7116_insert__UNIV,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_7117_perfect__space__class_OUNIV__not__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X: A] :
( ( top_top @ ( set @ A ) )
!= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% perfect_space_class.UNIV_not_singleton
thf(fact_7118_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_7119_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F2: B > A,A2: A,X: B] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F2 @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_7120_INF__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% INF_empty
thf(fact_7121_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y4: B] : C2
@ A3 ) )
= ( top_top @ A ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y4: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% INF_constant
thf(fact_7122_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) ) @ ( image @ B @ A @ F2 @ ( uminus_uminus @ ( set @ B ) @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_7123_INT__empty,axiom,
! [B: $tType,A: $tType,B3: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_7124_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( folding_insort_key @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ ( set @ A ) )
@ ^ [X2: A] : X2 ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_7125_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_7126_UNIV__nat__eq,axiom,
( ( top_top @ ( set @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UNIV_nat_eq
thf(fact_7127_INT__extend__simps_I3_J,axiom,
! [F: $tType,E4: $tType,C3: set @ E4,A3: E4 > ( set @ F ),B3: set @ F] :
( ( ( C3
= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E4 @ ( set @ F ) @ A3 @ C3 ) ) @ B3 )
= ( minus_minus @ ( set @ F ) @ ( top_top @ ( set @ F ) ) @ B3 ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E4 @ ( set @ F ) @ A3 @ C3 ) ) @ B3 )
= ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E4 @ ( set @ F )
@ ^ [X2: E4] : ( minus_minus @ ( set @ F ) @ ( A3 @ X2 ) @ B3 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_7128_bij__image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F2: A > B,B3: C > ( set @ A ),A3: set @ C] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ C @ ( set @ A ) @ B3 @ A3 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image @ C @ ( set @ B )
@ ^ [X2: C] : ( image @ A @ B @ F2 @ ( B3 @ X2 ) )
@ A3 ) ) ) ) ).
% bij_image_INT
thf(fact_7129_UN__UN__finite__eq,axiom,
! [A: $tType,A3: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [N3: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_7130_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F2: B > A] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_7131_UN__finite__subset,axiom,
! [A: $tType,A3: nat > ( set @ A ),C3: set @ A] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) @ C3 ) ) ).
% UN_finite_subset
thf(fact_7132_UN__finite2__eq,axiom,
! [A: $tType,A3: nat > ( set @ A ),B3: nat > ( set @ A ),K: nat] :
( ! [N2: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N2 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_7133_suminf__eq__SUP,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( canoni5634975068530333245id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ( ( suminf @ A )
= ( ^ [F3: nat > A] :
( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N3 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% suminf_eq_SUP
thf(fact_7134_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image @ nat @ nat
@ ^ [M3: nat] : ( modulo_modulo @ nat @ M3 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_7135_UN__finite2__subset,axiom,
! [A: $tType,A3: nat > ( set @ A ),B3: nat > ( set @ A ),K: nat] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N2 @ K ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_7136_UN__constant__eq,axiom,
! [A: $tType,B: $tType,A2: A,A3: set @ A,F2: A > ( set @ B ),C2: set @ B] :
( ( member @ A @ A2 @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( F2 @ X3 )
= C2 ) )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ A3 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_7137_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F2: nat > ( set @ A ),S2: set @ A] :
( ! [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ I2 ) @ S2 )
=> ( ( finite_finite2 @ A @ S2 )
=> ( ? [N5: nat] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ N5 )
=> ! [M4: nat] :
( ( ord_less_eq @ nat @ M4 @ N5 )
=> ( ( ord_less @ nat @ M4 @ N2 )
=> ( ord_less @ ( set @ A ) @ ( F2 @ M4 ) @ ( F2 @ N2 ) ) ) ) )
& ! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ( F2 @ N5 )
= ( F2 @ N2 ) ) ) )
=> ( ( F2 @ ( finite_card @ A @ S2 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_7138_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F3: A > nat,R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y4: A] :
( ( ord_less @ nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) )
| ( ( ord_less_eq @ nat @ ( F3 @ X2 ) @ ( F3 @ Y4 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_7139_Id__on__fold,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( id_on @ A @ A3 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X2: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A3 ) ) ) ).
% Id_on_fold
thf(fact_7140_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
( ( P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B5: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B5: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_7141_card__UNIV__bool,axiom,
( ( finite_card @ $o @ ( top_top @ ( set @ $o ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% card_UNIV_bool
thf(fact_7142_range__mult,axiom,
! [A2: real] :
( ( ( A2
= ( zero_zero @ real ) )
=> ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) )
= ( insert @ real @ ( zero_zero @ real ) @ ( bot_bot @ ( set @ real ) ) ) ) )
& ( ( A2
!= ( zero_zero @ real ) )
=> ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) )
= ( top_top @ ( set @ real ) ) ) ) ) ).
% range_mult
thf(fact_7143_INF__UNIV__bool__expand,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: $o > A] :
( ( complete_Inf_Inf @ A @ ( image @ $o @ A @ A3 @ ( top_top @ ( set @ $o ) ) ) )
= ( inf_inf @ A @ ( A3 @ $true ) @ ( A3 @ $false ) ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_7144_SUP__UNIV__bool__expand,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: $o > A] :
( ( complete_Sup_Sup @ A @ ( image @ $o @ A @ A3 @ ( top_top @ ( set @ $o ) ) ) )
= ( sup_sup @ A @ ( A3 @ $true ) @ ( A3 @ $false ) ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_7145_Un__eq__UN,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B4: set @ A] :
( complete_Sup_Sup @ ( set @ A )
@ ( image @ $o @ ( set @ A )
@ ^ [B5: $o] : ( if @ ( set @ A ) @ B5 @ A4 @ B4 )
@ ( top_top @ ( set @ $o ) ) ) ) ) ) ).
% Un_eq_UN
thf(fact_7146_UN__bool__eq,axiom,
! [A: $tType,A3: $o > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ $o @ ( set @ A ) @ A3 @ ( top_top @ ( set @ $o ) ) ) )
= ( sup_sup @ ( set @ A ) @ ( A3 @ $true ) @ ( A3 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_7147_INT__bool__eq,axiom,
! [A: $tType,A3: $o > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ $o @ ( set @ A ) @ A3 @ ( top_top @ ( set @ $o ) ) ) )
= ( inf_inf @ ( set @ A ) @ ( A3 @ $true ) @ ( A3 @ $false ) ) ) ).
% INT_bool_eq
thf(fact_7148_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X2: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_7149_infinite__UNIV__int,axiom,
~ ( finite_finite2 @ int @ ( top_top @ ( set @ int ) ) ) ).
% infinite_UNIV_int
thf(fact_7150_infinite__UNIV__listI,axiom,
! [A: $tType] :
~ ( finite_finite2 @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% infinite_UNIV_listI
thf(fact_7151_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_7152_Id__on__def_H,axiom,
! [A: $tType,A3: A > $o] :
( ( id_on @ A @ ( collect @ A @ A3 ) )
= ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y4: A] :
( ( X2 = Y4 )
& ( A3 @ X2 ) ) ) ) ) ).
% Id_on_def'
thf(fact_7153_Ints__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_Ints @ A )
= ( image @ int @ A @ ( ring_1_of_int @ A ) @ ( top_top @ ( set @ int ) ) ) ) ) ).
% Ints_def
thf(fact_7154_int__in__range__abs,axiom,
! [N: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N ) @ ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ).
% int_in_range_abs
thf(fact_7155_mlex__less,axiom,
! [A: $tType,F2: A > nat,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F2 @ R2 ) ) ) ).
% mlex_less
thf(fact_7156_mlex__iff,axiom,
! [A: $tType,X: A,Y: A,F2: A > nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F2 @ R2 ) )
= ( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y ) )
| ( ( ( F2 @ X )
= ( F2 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% mlex_iff
thf(fact_7157_mlex__leq,axiom,
! [A: $tType,F2: A > nat,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F2 @ X ) @ ( F2 @ 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 @ F2 @ R2 ) ) ) ) ).
% mlex_leq
thf(fact_7158_Id__on__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( id_on @ A @ ( set2 @ A @ Xs ) )
= ( set2 @ ( product_prod @ A @ A )
@ ( map @ A @ ( product_prod @ A @ A )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_7159_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A4: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X2: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A4 ) ) ) ) ).
% Id_on_def
thf(fact_7160_root__def,axiom,
( root
= ( ^ [N3: nat,X2: real] :
( if @ real
@ ( N3
= ( 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 ) @ N3 ) )
@ X2 ) ) ) ) ).
% root_def
thf(fact_7161_card__UNIV__char,axiom,
( ( finite_card @ char @ ( top_top @ ( set @ char ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% card_UNIV_char
thf(fact_7162_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_7163_top2I,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : ( top_top @ ( A > B > $o ) @ X @ Y ) ).
% top2I
thf(fact_7164_UNIV__bool,axiom,
( ( top_top @ ( set @ $o ) )
= ( insert @ $o @ $false @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% UNIV_bool
thf(fact_7165_the__inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( the_inv_into @ A @ B )
= ( ^ [A4: set @ A,F3: A > B,X2: B] :
( the @ A
@ ^ [Y4: A] :
( ( member @ A @ Y4 @ A4 )
& ( ( F3 @ Y4 )
= X2 ) ) ) ) ) ).
% the_inv_into_def
thf(fact_7166_UNIV__char__of__nat,axiom,
( ( top_top @ ( set @ char ) )
= ( image @ 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_7167_char__of__mod__256,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: A] :
( ( unique5772411509450598832har_of @ A @ ( modulo_modulo @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
= ( unique5772411509450598832har_of @ A @ N ) ) ) ).
% char_of_mod_256
thf(fact_7168_char__of__nat,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( unique5772411509450598832har_of @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( unique5772411509450598832har_of @ nat @ N ) ) ) ).
% char_of_nat
thf(fact_7169_char__of__quasi__inj,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: A,N: A] :
( ( ( unique5772411509450598832har_of @ A @ M )
= ( unique5772411509450598832har_of @ A @ N ) )
= ( ( modulo_modulo @ A @ M @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) )
= ( modulo_modulo @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% char_of_quasi_inj
thf(fact_7170_char__of__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,M: A] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N )
=> ( ( unique5772411509450598832har_of @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ M ) )
= ( unique5772411509450598832har_of @ A @ M ) ) ) ) ).
% char_of_take_bit_eq
thf(fact_7171_of__char__of,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [A2: A] :
( ( comm_s6883823935334413003f_char @ A @ ( unique5772411509450598832har_of @ A @ A2 ) )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% of_char_of
thf(fact_7172_char__of__def,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( unique5772411509450598832har_of @ A )
= ( ^ [N3: A] :
( char2
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( one_one @ nat ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% char_of_def
thf(fact_7173_of__char__mod__256,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [C2: char] :
( ( modulo_modulo @ A @ ( comm_s6883823935334413003f_char @ A @ C2 ) @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) )
= ( comm_s6883823935334413003f_char @ A @ C2 ) ) ) ).
% of_char_mod_256
thf(fact_7174_of__char__Char,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( comm_s6883823935334413003f_char @ A @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cons @ $o @ B0 @ ( cons @ $o @ B1 @ ( cons @ $o @ B22 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) ) ) ) ) ).
% of_char_Char
thf(fact_7175_char_Osize_I2_J,axiom,
! [X15: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size @ char @ ( char2 @ X15 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size(2)
thf(fact_7176_nat__of__char__less__256,axiom,
! [C2: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C2 ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% nat_of_char_less_256
thf(fact_7177_range__nat__of__char,axiom,
( ( image @ 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_7178_char__of__eq__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: A,C2: char] :
( ( ( unique5772411509450598832har_of @ A @ N )
= C2 )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N )
= ( comm_s6883823935334413003f_char @ A @ C2 ) ) ) ) ).
% char_of_eq_iff
thf(fact_7179_integer__of__char__code,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( zero_neq_one_of_bool @ code_integer @ B72 ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B62 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B52 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B42 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B32 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B22 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B1 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B0 ) ) ) ).
% integer_of_char_code
thf(fact_7180_char_Osize__gen,axiom,
! [X15: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_char @ ( char2 @ X15 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size_gen
thf(fact_7181_char__of__integer__code,axiom,
( char_of_integer
= ( ^ [K3: code_integer] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q0: code_integer,B02: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q1: code_integer,B15: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q22: code_integer,B23: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q32: code_integer,B33: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q42: code_integer,B43: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q52: code_integer,B53: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q62: code_integer,B63: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Uu: code_integer] : ( char2 @ B02 @ B15 @ B23 @ B33 @ B43 @ B53 @ B63 )
@ ( code_bit_cut_integer @ Q62 ) )
@ ( code_bit_cut_integer @ Q52 ) )
@ ( code_bit_cut_integer @ Q42 ) )
@ ( code_bit_cut_integer @ Q32 ) )
@ ( code_bit_cut_integer @ Q22 ) )
@ ( code_bit_cut_integer @ Q1 ) )
@ ( code_bit_cut_integer @ Q0 ) )
@ ( code_bit_cut_integer @ K3 ) ) ) ) ).
% char_of_integer_code
thf(fact_7182_String_Ochar__of__ascii__of,axiom,
! [C2: char] :
( ( comm_s6883823935334413003f_char @ nat @ ( ascii_of @ C2 ) )
= ( bit_se2584673776208193580ke_bit @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( comm_s6883823935334413003f_char @ nat @ C2 ) ) ) ).
% String.char_of_ascii_of
thf(fact_7183_Gcd__eq__Max,axiom,
! [M7: set @ nat] :
( ( finite_finite2 @ nat @ M7 )
=> ( ( M7
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( gcd_Gcd @ nat @ M7 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image @ nat @ ( set @ nat )
@ ^ [M3: nat] :
( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M3 ) )
@ M7 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_7184_INF__filter__not__bot,axiom,
! [I5: $tType,A: $tType,B3: set @ I5,F4: I5 > ( filter @ A )] :
( ! [X10: set @ I5] :
( ( ord_less_eq @ ( set @ I5 ) @ X10 @ B3 )
=> ( ( finite_finite2 @ I5 @ X10 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I5 @ ( filter @ A ) @ F4 @ X10 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I5 @ ( filter @ A ) @ F4 @ B3 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_7185_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_7186_Gcd__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Gcd_UNIV
thf(fact_7187_Gcd__insert,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A2: A,A3: set @ A] :
( ( gcd_Gcd @ A @ ( insert @ A @ A2 @ A3 ) )
= ( gcd_gcd @ A @ A2 @ ( gcd_Gcd @ A @ A3 ) ) ) ) ).
% Gcd_insert
thf(fact_7188_Gcd__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_Gcd @ A @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% Gcd_2
thf(fact_7189_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A] :
( ( ( gcd_Gcd @ A @ A3 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_7190_Gcd__eq__1__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A2: A,A3: set @ A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( member @ A @ A2 @ A3 )
=> ( ( gcd_Gcd @ A @ A3 )
= ( one_one @ A ) ) ) ) ) ).
% Gcd_eq_1_I
thf(fact_7191_Gcd__1,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A] :
( ( member @ A @ ( one_one @ A ) @ A3 )
=> ( ( gcd_Gcd @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% Gcd_1
thf(fact_7192_Gcd__nat__eq__one,axiom,
! [N4: set @ nat] :
( ( member @ nat @ ( one_one @ nat ) @ N4 )
=> ( ( gcd_Gcd @ nat @ N4 )
= ( one_one @ nat ) ) ) ).
% Gcd_nat_eq_one
thf(fact_7193_Gcd__dvd__nat,axiom,
! [A2: nat,A3: set @ nat] :
( ( member @ nat @ A2 @ A3 )
=> ( dvd_dvd @ nat @ ( gcd_Gcd @ nat @ A3 ) @ A2 ) ) ).
% Gcd_dvd_nat
thf(fact_7194_Gcd__greatest__nat,axiom,
! [A3: set @ nat,A2: nat] :
( ! [B6: nat] :
( ( member @ nat @ B6 @ A3 )
=> ( dvd_dvd @ nat @ A2 @ B6 ) )
=> ( dvd_dvd @ nat @ A2 @ ( gcd_Gcd @ nat @ A3 ) ) ) ).
% Gcd_greatest_nat
thf(fact_7195_Gcd__dvd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( dvd_dvd @ A @ ( gcd_Gcd @ A @ A3 ) @ A2 ) ) ) ).
% Gcd_dvd
thf(fact_7196_dvd__GcdD,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [X: A,A3: set @ A,Y: A] :
( ( dvd_dvd @ A @ X @ ( gcd_Gcd @ A @ A3 ) )
=> ( ( member @ A @ Y @ A3 )
=> ( dvd_dvd @ A @ X @ Y ) ) ) ) ).
% dvd_GcdD
thf(fact_7197_dvd__Gcd__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [X: A,A3: set @ A] :
( ( dvd_dvd @ A @ X @ ( gcd_Gcd @ A @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( dvd_dvd @ A @ X @ X2 ) ) ) ) ) ).
% dvd_Gcd_iff
thf(fact_7198_Gcd__greatest,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A,A2: A] :
( ! [B6: A] :
( ( member @ A @ B6 @ A3 )
=> ( dvd_dvd @ A @ A2 @ B6 ) )
=> ( dvd_dvd @ A @ A2 @ ( gcd_Gcd @ A @ A3 ) ) ) ) ).
% Gcd_greatest
thf(fact_7199_Gcd__in,axiom,
! [A3: set @ nat] :
( ! [A6: nat,B6: nat] :
( ( member @ nat @ A6 @ A3 )
=> ( ( member @ nat @ B6 @ A3 )
=> ( member @ nat @ ( gcd_gcd @ nat @ A6 @ B6 ) @ A3 ) ) )
=> ( ( A3
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( gcd_Gcd @ nat @ A3 ) @ A3 ) ) ) ).
% Gcd_in
thf(fact_7200_Inf__filter__not__bot,axiom,
! [A: $tType,B3: set @ ( filter @ A )] :
( ! [X10: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X10 @ B3 )
=> ( ( finite_finite2 @ ( filter @ A ) @ X10 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X10 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B3 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_7201_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I6: set @ A,F4: A > ( filter @ B )] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I6 )
=> ? [X6: A] :
( ( member @ A @ X6 @ I6 )
& ( ord_less_eq @ ( filter @ B ) @ ( F4 @ X6 ) @ ( inf_inf @ ( filter @ B ) @ ( F4 @ I2 ) @ ( F4 @ J2 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F4 @ I6 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ I6 )
& ( ( F4 @ X2 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_7202_Gcd__mono,axiom,
! [A: $tType,B: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ B,F2: B > A,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( dvd_dvd @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( dvd_dvd @ A @ ( gcd_Gcd @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( gcd_Gcd @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ).
% Gcd_mono
thf(fact_7203_Gcd__remove0__nat,axiom,
! [M7: set @ nat] :
( ( finite_finite2 @ nat @ M7 )
=> ( ( gcd_Gcd @ nat @ M7 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M7 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_7204_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_7205_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D: 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 )
=> ( D
= ( 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 ) )
=> ( D
= ( 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 )
=> ( D
= ( 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 ) @ D @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_7206_abs__Gcd__eq,axiom,
! [K6: set @ int] :
( ( abs_abs @ int @ ( gcd_Gcd @ int @ K6 ) )
= ( gcd_Gcd @ int @ K6 ) ) ).
% abs_Gcd_eq
thf(fact_7207_Gcd__abs__eq,axiom,
! [K6: set @ int] :
( ( gcd_Gcd @ int @ ( image @ int @ int @ ( abs_abs @ int ) @ K6 ) )
= ( gcd_Gcd @ int @ K6 ) ) ).
% Gcd_abs_eq
thf(fact_7208_Gcd__nat__abs__eq,axiom,
! [K6: set @ int] :
( ( gcd_Gcd @ nat
@ ( image @ int @ nat
@ ^ [K3: int] : ( nat2 @ ( abs_abs @ int @ K3 ) )
@ K6 ) )
= ( nat2 @ ( gcd_Gcd @ int @ K6 ) ) ) ).
% Gcd_nat_abs_eq
thf(fact_7209_Gcd__int__eq,axiom,
! [N4: set @ nat] :
( ( gcd_Gcd @ int @ ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ N4 ) )
= ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ N4 ) ) ) ).
% Gcd_int_eq
thf(fact_7210_less__filter__def,axiom,
! [A: $tType] :
( ( ord_less @ ( filter @ A ) )
= ( ^ [F9: filter @ A,F10: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F9 @ F10 )
& ~ ( ord_less_eq @ ( filter @ A ) @ F10 @ F9 ) ) ) ) ).
% less_filter_def
thf(fact_7211_Gcd__int__greater__eq__0,axiom,
! [K6: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Gcd @ int @ K6 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_7212_Gcd__dvd__int,axiom,
! [A2: int,A3: set @ int] :
( ( member @ int @ A2 @ A3 )
=> ( dvd_dvd @ int @ ( gcd_Gcd @ int @ A3 ) @ A2 ) ) ).
% Gcd_dvd_int
thf(fact_7213_Gcd__greatest__int,axiom,
! [A3: set @ int,A2: int] :
( ! [B6: int] :
( ( member @ int @ B6 @ A3 )
=> ( dvd_dvd @ int @ A2 @ B6 ) )
=> ( dvd_dvd @ int @ A2 @ ( gcd_Gcd @ int @ A3 ) ) ) ).
% Gcd_greatest_int
thf(fact_7214_DERIV__at__within__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,Z3: A,X: A,S2: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z3 @ X ) @ ( image @ A @ A @ ( plus_plus @ A @ Z3 ) @ S2 ) ) )
= ( has_field_derivative @ A
@ ^ [X2: A] : ( F2 @ ( plus_plus @ A @ Z3 @ X2 ) )
@ Y
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_at_within_shift
thf(fact_7215_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( power_power @ A @ ( F2 @ X2 ) @ ( suc @ N ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) @ ( times_times @ A @ D @ ( power_power @ A @ ( F2 @ X ) @ N ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power_Suc
thf(fact_7216_DERIV__inverse,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,S: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_inverse
thf(fact_7217_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( power_power @ A @ ( F2 @ X2 ) @ N )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( times_times @ A @ D @ ( power_power @ A @ ( F2 @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power
thf(fact_7218_DERIV__mult_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X ) @ E5 ) @ ( times_times @ A @ D @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_mult'
thf(fact_7219_DERIV__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,X: A,S: set @ A,G: A > A,Db: A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G @ X ) ) @ ( times_times @ A @ Db @ ( F2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_mult
thf(fact_7220_DERIV__cmult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A,C2: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( times_times @ A @ C2 @ D )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_cmult
thf(fact_7221_DERIV__cmult__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A,C2: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( times_times @ A @ D @ C2 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_cmult_right
thf(fact_7222_has__field__derivative__cosh,axiom,
! [A17: $tType] :
( ( ( real_Vector_banach @ A17 )
& ( real_V3459762299906320749_field @ A17 ) )
=> ! [G: A17 > A17,Db: A17,X: A17,S: set @ A17] :
( ( has_field_derivative @ A17 @ G @ Db @ ( topolo174197925503356063within @ A17 @ X @ S ) )
=> ( has_field_derivative @ A17
@ ^ [X2: A17] : ( cosh @ A17 @ ( G @ X2 ) )
@ ( times_times @ A17 @ ( sinh @ A17 @ ( G @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A17 @ X @ S ) ) ) ) ).
% has_field_derivative_cosh
thf(fact_7223_has__field__derivative__sinh,axiom,
! [A17: $tType] :
( ( ( real_Vector_banach @ A17 )
& ( real_V3459762299906320749_field @ A17 ) )
=> ! [G: A17 > A17,Db: A17,X: A17,S: set @ A17] :
( ( has_field_derivative @ A17 @ G @ Db @ ( topolo174197925503356063within @ A17 @ X @ S ) )
=> ( has_field_derivative @ A17
@ ^ [X2: A17] : ( sinh @ A17 @ ( G @ X2 ) )
@ ( times_times @ A17 @ ( cosh @ A17 @ ( G @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A17 @ X @ S ) ) ) ) ).
% has_field_derivative_sinh
thf(fact_7224_DERIV__ident,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F4: filter @ A] :
( has_field_derivative @ A
@ ^ [X2: A] : X2
@ ( one_one @ A )
@ F4 ) ) ).
% DERIV_ident
thf(fact_7225_field__differentiable__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,F4: filter @ A,G: A > A,G4: A] :
( ( has_field_derivative @ A @ F2 @ F7 @ F4 )
=> ( ( has_field_derivative @ A @ G @ G4 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z7: A] : ( plus_plus @ A @ ( F2 @ Z7 ) @ ( G @ Z7 ) )
@ ( plus_plus @ A @ F7 @ G4 )
@ F4 ) ) ) ) ).
% field_differentiable_add
thf(fact_7226_DERIV__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( plus_plus @ A @ D @ E5 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_add
thf(fact_7227_DERIV__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( minus_minus @ A @ D @ E5 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_diff
thf(fact_7228_field__differentiable__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,F4: filter @ A,G: A > A,G4: A] :
( ( has_field_derivative @ A @ F2 @ F7 @ F4 )
=> ( ( has_field_derivative @ A @ G @ G4 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z7: A] : ( minus_minus @ A @ ( F2 @ Z7 ) @ ( G @ Z7 ) )
@ ( minus_minus @ A @ F7 @ G4 )
@ F4 ) ) ) ) ).
% field_differentiable_diff
thf(fact_7229_at__le,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ord_less_eq @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X @ S ) @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ).
% at_le
thf(fact_7230_has__field__derivative__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,X: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% has_field_derivative_subset
thf(fact_7231_DERIV__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,X: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% DERIV_subset
thf(fact_7232_has__field__derivative__scaleR__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,F4: filter @ A,C2: real] :
( ( has_field_derivative @ A @ F2 @ D @ F4 )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ C2 @ ( F2 @ X2 ) )
@ ( real_V8093663219630862766scaleR @ A @ C2 @ D )
@ F4 ) ) ) ).
% has_field_derivative_scaleR_right
thf(fact_7233_DERIV__cdivide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A,C2: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F2 @ X2 ) @ C2 )
@ ( divide_divide @ A @ D @ C2 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_cdivide
thf(fact_7234_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K: A,F4: filter @ A] :
( has_field_derivative @ A
@ ^ [X2: A] : K
@ ( zero_zero @ A )
@ F4 ) ) ).
% DERIV_const
thf(fact_7235_field__differentiable__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,F4: filter @ A] :
( ( has_field_derivative @ A @ F2 @ F7 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z7: A] : ( uminus_uminus @ A @ ( F2 @ Z7 ) )
@ ( uminus_uminus @ A @ F7 )
@ F4 ) ) ) ).
% field_differentiable_minus
thf(fact_7236_DERIV__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( uminus_uminus @ A @ ( F2 @ X2 ) )
@ ( uminus_uminus @ A @ D )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_minus
thf(fact_7237_has__real__derivative__neg__dec__right,axiom,
! [F2: real > real,L: real,X: real,S2: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ S2 ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H6 ) @ S2 )
=> ( ( ord_less @ real @ H6 @ D6 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X @ H6 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_7238_has__real__derivative__pos__inc__right,axiom,
! [F2: real > real,L: real,X: real,S2: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H6 ) @ S2 )
=> ( ( ord_less @ real @ H6 @ D6 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( plus_plus @ real @ X @ H6 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_7239_has__real__derivative__neg__dec__left,axiom,
! [F2: real > real,L: real,X: real,S2: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ S2 ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H6 ) @ S2 )
=> ( ( ord_less @ real @ H6 @ D6 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( minus_minus @ real @ X @ H6 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_7240_has__real__derivative__pos__inc__left,axiom,
! [F2: real > real,L: real,X: real,S2: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H6 ) @ S2 )
=> ( ( ord_less @ real @ H6 @ D6 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X @ H6 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_7241_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X ) ) @ D ) @ ( inverse_inverse @ A @ ( F2 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_7242_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D @ ( G @ X ) ) @ ( times_times @ A @ ( F2 @ X ) @ E5 ) ) @ ( times_times @ A @ ( G @ X ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_7243_DERIV__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ B )
=> ! [S2: set @ A,F2: B > A > B,F7: C > A > B,X: C,F4: filter @ B] :
( ! [N2: A] :
( ( member @ A @ N2 @ S2 )
=> ( has_field_derivative @ B
@ ^ [X2: B] : ( F2 @ X2 @ N2 )
@ ( F7 @ X @ N2 )
@ F4 ) )
=> ( has_field_derivative @ B
@ ^ [X2: B] : ( groups7311177749621191930dd_sum @ A @ B @ ( F2 @ X2 ) @ S2 )
@ ( groups7311177749621191930dd_sum @ A @ B @ ( F7 @ X ) @ S2 )
@ F4 ) ) ) ).
% DERIV_sum
thf(fact_7244_DERIV__fun__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( cos @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( uminus_uminus @ A @ ( sin @ A @ ( G @ X ) ) ) @ M )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_cos
thf(fact_7245_DERIV__cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K: A,Xa: A] :
( has_field_derivative @ A
@ ^ [X2: A] : ( cos @ A @ ( plus_plus @ A @ X2 @ K ) )
@ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa @ K ) ) )
@ ( topolo174197925503356063within @ A @ Xa @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_cos_add
thf(fact_7246_DERIV__pos__inc__left,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( ord_less @ real @ H6 @ D6 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X @ H6 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_7247_DERIV__neg__dec__left,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( ord_less @ real @ H6 @ D6 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( minus_minus @ real @ X @ H6 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_7248_DERIV__neg__dec__right,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( ord_less @ real @ H6 @ D6 )
=> ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X @ H6 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_7249_DERIV__pos__inc__right,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [H6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H6 )
=> ( ( ord_less @ real @ H6 @ D6 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( plus_plus @ real @ X @ H6 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_7250_DERIV__ln__divide,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln_divide
thf(fact_7251_DERIV__const__average,axiom,
! [A2: real,B2: real,V2: real > real,K: real] :
( ( A2 != B2 )
=> ( ! [X3: real] : ( has_field_derivative @ real @ V2 @ K @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( V2 @ ( divide_divide @ real @ ( plus_plus @ real @ A2 @ B2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ real @ ( plus_plus @ real @ ( V2 @ A2 ) @ ( V2 @ B2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% DERIV_const_average
thf(fact_7252_deriv__nonneg__imp__mono,axiom,
! [A2: real,B2: real,G: real > real,G4: real > real] :
( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( has_field_derivative @ real @ G @ ( G4 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G4 @ X3 ) ) )
=> ( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ord_less_eq @ real @ ( G @ A2 ) @ ( G @ B2 ) ) ) ) ) ).
% deriv_nonneg_imp_mono
thf(fact_7253_DERIV__pos__imp__increasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y2: real] :
( ( has_field_derivative @ real @ F2 @ Y2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y2 ) ) ) )
=> ( ord_less @ real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_7254_DERIV__neg__imp__decreasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y2: real] :
( ( has_field_derivative @ real @ F2 @ Y2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y2 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_7255_DERIV__nonpos__imp__nonincreasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y2: real] :
( ( has_field_derivative @ real @ F2 @ Y2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ Y2 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less_eq @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ).
% DERIV_nonpos_imp_nonincreasing
thf(fact_7256_DERIV__nonneg__imp__nondecreasing,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y2: real] :
( ( has_field_derivative @ real @ F2 @ Y2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 ) ) ) )
=> ( ord_less_eq @ real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ).
% DERIV_nonneg_imp_nondecreasing
thf(fact_7257_MVT2,axiom,
! [A2: real,B2: real,F2: real > real,F7: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( F7 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z: real] :
( ( ord_less @ real @ A2 @ Z )
& ( ord_less @ real @ Z @ B2 )
& ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( F7 @ Z ) ) ) ) ) ) ).
% MVT2
thf(fact_7258_DERIV__ln,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( inverse_inverse @ real @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln
thf(fact_7259_DERIV__isconst__all,axiom,
! [F2: real > real,X: real,Y: real] :
( ! [X3: real] : ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( F2 @ X )
= ( F2 @ Y ) ) ) ).
% DERIV_isconst_all
thf(fact_7260_DERIV__mirror,axiom,
! [F2: real > real,Y: real,X: real] :
( ( has_field_derivative @ real @ F2 @ Y @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ X ) @ ( top_top @ ( set @ real ) ) ) )
= ( has_field_derivative @ real
@ ^ [X2: real] : ( F2 @ ( uminus_uminus @ real @ X2 ) )
@ ( uminus_uminus @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_mirror
thf(fact_7261_DERIV__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Y: A,X: A,Z3: A] :
( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A
@ ^ [X2: A] : ( F2 @ ( plus_plus @ A @ X2 @ Z3 ) )
@ Y
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_shift
thf(fact_7262_DERIV__fun__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( exp @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( exp @ A @ ( G @ X ) ) @ M )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_exp
thf(fact_7263_DERIV__fun__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( sin @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( cos @ A @ ( G @ X ) ) @ M )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_sin
thf(fact_7264_DERIV__chain__s,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [S: set @ A,G: A > A,G4: A > A,F2: A > A,F7: A,X: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( has_field_derivative @ A @ G @ ( G4 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ A @ ( F2 @ X ) @ S )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( times_times @ A @ F7 @ ( G4 @ ( F2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% DERIV_chain_s
thf(fact_7265_DERIV__chain3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [G: A > A,G4: A > A,F2: A > A,F7: A,X: A] :
( ! [X3: A] : ( has_field_derivative @ A @ G @ ( G4 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( times_times @ A @ F7 @ ( G4 @ ( F2 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% DERIV_chain3
thf(fact_7266_DERIV__chain2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Da: A,G: A > A,X: A,Db: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ ( G @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( F2 @ ( G @ X2 ) )
@ ( times_times @ A @ Da @ Db )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_chain2
thf(fact_7267_DERIV__chain_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ ( F2 @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( times_times @ A @ E5 @ D )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_chain'
thf(fact_7268_DERIV__local__min,axiom,
! [F2: real > real,L: real,X: real,D4: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D4 )
=> ( ord_less_eq @ real @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_min
thf(fact_7269_DERIV__local__max,axiom,
! [F2: real > real,L: real,X: real,D4: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D4 )
=> ( ord_less_eq @ real @ ( F2 @ Y3 ) @ ( F2 @ X ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_max
thf(fact_7270_DERIV__local__const,axiom,
! [F2: real > real,L: real,X: real,D4: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D4 )
=> ( ( F2 @ X )
= ( F2 @ Y3 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_const
thf(fact_7271_DERIV__pow,axiom,
! [N: nat,X: real,S: set @ real] :
( has_field_derivative @ real
@ ^ [X2: real] : ( power_power @ real @ X2 @ N )
@ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ X @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ S ) ) ).
% DERIV_pow
thf(fact_7272_termdiffs__strong__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [Y3: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Y3 @ N3 ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% termdiffs_strong_converges_everywhere
thf(fact_7273_at__within__Icc__at,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less @ A @ A2 @ X )
=> ( ( ord_less @ A @ X @ B2 )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% at_within_Icc_at
thf(fact_7274_DERIV__fun__pow,axiom,
! [G: real > real,M: real,X: real,N: nat] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( power_power @ real @ ( G @ X2 ) @ N )
@ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( G @ X ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_fun_pow
thf(fact_7275_at__within__Icc__at__left,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( topolo174197925503356063within @ A @ B2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ).
% at_within_Icc_at_left
thf(fact_7276_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A,G: A > A,E2: A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y4: A] : ( divide_divide @ A @ ( F2 @ Y4 ) @ ( G @ Y4 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D4 @ ( G @ X ) ) @ ( times_times @ A @ E2 @ ( F2 @ X ) ) ) @ ( power_power @ A @ ( G @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_7277_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D4: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D4 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F2 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_7278_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K6: real,C2: nat > A,F2: A > A,F7: A,Z3: A] :
( ! [Z: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) )
@ ( F2 @ Z ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ Z3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K6 )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) )
@ F7 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_7279_has__real__derivative__powr,axiom,
! [Z3: real,R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z3 )
=> ( has_field_derivative @ real
@ ^ [Z7: real] : ( powr @ real @ Z7 @ 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_7280_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_7281_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ X @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_7282_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K6: real,C2: nat > A,Z3: A] :
( ! [Z: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K6 )
=> ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Z @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K6 )
=> ( has_field_derivative @ A
@ ^ [Z7: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Z7 @ N3 ) ) )
@ ( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N3 ) @ ( power_power @ A @ Z3 @ N3 ) ) )
@ ( topolo174197925503356063within @ A @ Z3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_7283_DERIV__fun__powr,axiom,
! [G: real > real,M: real,X: real,R3: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( powr @ real @ ( G @ X2 ) @ R3 )
@ ( times_times @ real @ ( times_times @ real @ R3 @ ( powr @ real @ ( G @ X ) @ ( minus_minus @ real @ R3 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_fun_powr
thf(fact_7284_DERIV__log,axiom,
! [X: real,B2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( log @ B2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B2 ) @ X ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_log
thf(fact_7285_DERIV__powr,axiom,
! [G: real > real,M: real,X: real,F2: real > real,R3: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_field_derivative @ real @ F2 @ R3 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( powr @ real @ ( G @ X2 ) @ ( F2 @ X2 ) )
@ ( times_times @ real @ ( powr @ real @ ( G @ X ) @ ( F2 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ R3 @ ( ln_ln @ real @ ( G @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M @ ( F2 @ X ) ) @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_powr
thf(fact_7286_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_7287_DERIV__real__sqrt,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ sqrt @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_real_sqrt
thf(fact_7288_DERIV__arctan,axiom,
! [X: real] : ( has_field_derivative @ real @ arctan @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ).
% DERIV_arctan
thf(fact_7289_arsinh__real__has__field__derivative,axiom,
! [X: real,A3: set @ real] : ( has_field_derivative @ real @ ( arsinh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A3 ) ) ).
% arsinh_real_has_field_derivative
thf(fact_7290_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_7291_has__field__derivative__tanh,axiom,
! [A17: $tType] :
( ( ( real_Vector_banach @ A17 )
& ( real_V3459762299906320749_field @ A17 ) )
=> ! [G: A17 > A17,X: A17,Db: A17,S: set @ A17] :
( ( ( cosh @ A17 @ ( G @ X ) )
!= ( zero_zero @ A17 ) )
=> ( ( has_field_derivative @ A17 @ G @ Db @ ( topolo174197925503356063within @ A17 @ X @ S ) )
=> ( has_field_derivative @ A17
@ ^ [X2: A17] : ( tanh @ A17 @ ( G @ X2 ) )
@ ( times_times @ A17 @ ( minus_minus @ A17 @ ( one_one @ A17 ) @ ( power_power @ A17 @ ( tanh @ A17 @ ( G @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A17 @ X @ S ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_7292_DERIV__real__sqrt__generic,axiom,
! [X: real,D: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D
= ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( has_field_derivative @ real @ sqrt @ D @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_7293_arcosh__real__has__field__derivative,axiom,
! [X: real,A3: set @ real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( has_field_derivative @ real @ ( arcosh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A3 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_7294_artanh__real__has__field__derivative,axiom,
! [X: real,A3: set @ real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ ( artanh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A3 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_7295_Gcd__int__def,axiom,
( ( gcd_Gcd @ int )
= ( ^ [K7: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K7 ) ) ) ) ) ).
% Gcd_int_def
thf(fact_7296_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_7297_DERIV__arccos,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arccos @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arccos
thf(fact_7298_DERIV__arcsin,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arcsin @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arcsin
thf(fact_7299_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F2: real > real,X: real,N: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_7300_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F2: real > real,X: real,N: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
& ! [M4: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_7301_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_7302_Maclaurin__minus,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ H2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ H2 @ T4 )
& ( ord_less_eq @ real @ T4 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less @ real @ H2 @ T4 )
& ( ord_less @ real @ T4 @ ( zero_zero @ real ) )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ H2 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_7303_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F2: real > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ H2 )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ H2 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_7304_Maclaurin,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less @ real @ T4 @ H2 )
& ( ( F2 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ H2 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_7305_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F2: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ! [M4: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T4 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_7306_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F2: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_7307_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ A2 @ T4 )
& ( ord_less_eq @ real @ T4 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A2 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ? [T4: real] :
( ( ord_less @ real @ A2 @ T4 )
& ( ord_less @ real @ T4 @ C2 )
& ( ( F2 @ A2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ C2 ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_7308_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ A2 @ T4 )
& ( ord_less_eq @ real @ T4 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C2 )
=> ( ( ord_less @ real @ C2 @ B2 )
=> ? [T4: real] :
( ( ord_less @ real @ C2 @ T4 )
& ( ord_less @ real @ T4 @ B2 )
& ( ( F2 @ B2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ C2 ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_7309_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ A2 @ T4 )
& ( ord_less_eq @ real @ T4 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ( ( ord_less_eq @ real @ A2 @ X )
=> ( ( ord_less_eq @ real @ X @ B2 )
=> ( ( X != C2 )
=> ? [T4: real] :
( ( ( ord_less @ real @ X @ C2 )
=> ( ( ord_less @ real @ X @ T4 )
& ( ord_less @ real @ T4 @ C2 ) ) )
& ( ~ ( ord_less @ real @ X @ C2 )
=> ( ( ord_less @ real @ C2 @ T4 )
& ( ord_less @ real @ T4 @ X ) ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ C2 ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C2 ) @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_7310_Maclaurin__lemma2,axiom,
! [N: nat,H2: real,Diff: nat > real > real,K: nat,B3: real] :
( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N
= ( suc @ K ) )
=> ! [M2: nat,T8: real] :
( ( ( ord_less @ nat @ M2 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H2 ) )
=> ( has_field_derivative @ real
@ ^ [U: real] :
( minus_minus @ real @ ( Diff @ M2 @ U )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M2 @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ U @ P6 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ M2 ) ) )
@ ( times_times @ real @ B3 @ ( divide_divide @ real @ ( power_power @ real @ U @ ( minus_minus @ nat @ N @ M2 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M2 ) @ T8 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M2 ) @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ T8 @ P6 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ M2 ) ) ) )
@ ( times_times @ real @ B3 @ ( divide_divide @ real @ ( power_power @ real @ T8 @ ( minus_minus @ nat @ N @ ( suc @ M2 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_7311_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_7312_DERIV__power__series_H,axiom,
! [R2: real,F2: nat > real,X0: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R2 ) @ R2 ) )
=> ( summable @ real
@ ^ [N3: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N3 ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) @ ( power_power @ real @ X3 @ N3 ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R2 ) @ R2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( has_field_derivative @ real
@ ^ [X2: real] :
( suminf @ real
@ ^ [N3: nat] : ( times_times @ real @ ( F2 @ N3 ) @ ( power_power @ real @ X2 @ ( suc @ N3 ) ) ) )
@ ( suminf @ real
@ ^ [N3: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N3 ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) @ ( power_power @ real @ X0 @ N3 ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_7313_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G4: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X ) )
=> ( ( ord_less @ real @ ( G @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arcsin @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_7314_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U2: A] :
( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U2 ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less @ A @ I @ U2 ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_7315_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or5935395276787703475ssThan @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanLessThan_empty
thf(fact_7316_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_7317_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_7318_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioo_iff
thf(fact_7319_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_7320_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanLessThan
thf(fact_7321_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_7322_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanLessThan
thf(fact_7323_has__derivative__in__compose,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: A > B,F7: A > B,X: A,S: set @ A,G: B > C,G4: B > C] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_derivative @ B @ C @ G @ G4 @ ( topolo174197925503356063within @ B @ ( F2 @ X ) @ ( image @ A @ B @ F2 @ S ) ) )
=> ( has_derivative @ A @ C
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ^ [X2: A] : ( G4 @ ( F7 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_in_compose
thf(fact_7324_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioo
thf(fact_7325_has__derivative__mult,axiom,
! [A: $tType,D3: $tType] :
( ( ( real_V822414075346904944vector @ D3 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F2: D3 > A,F7: D3 > A,X: D3,S: set @ D3,G: D3 > A,G4: D3 > A] :
( ( has_derivative @ D3 @ A @ F2 @ F7 @ ( topolo174197925503356063within @ D3 @ X @ S ) )
=> ( ( has_derivative @ D3 @ A @ G @ G4 @ ( topolo174197925503356063within @ D3 @ X @ S ) )
=> ( has_derivative @ D3 @ A
@ ^ [X2: D3] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [H: D3] : ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X ) @ ( G4 @ H ) ) @ ( times_times @ A @ ( F7 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ D3 @ X @ S ) ) ) ) ) ).
% has_derivative_mult
thf(fact_7326_has__derivative__scaleR,axiom,
! [C: $tType,D3: $tType] :
( ( ( real_V822414075346904944vector @ D3 )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: D3 > real,F7: D3 > real,X: D3,S: set @ D3,G: D3 > C,G4: D3 > C] :
( ( has_derivative @ D3 @ real @ F2 @ F7 @ ( topolo174197925503356063within @ D3 @ X @ S ) )
=> ( ( has_derivative @ D3 @ C @ G @ G4 @ ( topolo174197925503356063within @ D3 @ X @ S ) )
=> ( has_derivative @ D3 @ C
@ ^ [X2: D3] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [H: D3] : ( plus_plus @ C @ ( real_V8093663219630862766scaleR @ C @ ( F2 @ X ) @ ( G4 @ H ) ) @ ( real_V8093663219630862766scaleR @ C @ ( F7 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ D3 @ X @ S ) ) ) ) ) ).
% has_derivative_scaleR
thf(fact_7327_has__derivative__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X: A,S: set @ A,T2: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% has_derivative_subset
thf(fact_7328_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: A > B,X: A] :
( ( has_derivative @ A @ B
@ ^ [X2: A] : ( zero_zero @ B )
@ F4
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( F4
= ( ^ [H: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_7329_has__derivative__compose,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [F2: A > B,F7: A > B,X: A,S: set @ A,G: B > C,G4: B > C] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_derivative @ B @ C @ G @ G4 @ ( topolo174197925503356063within @ B @ ( F2 @ X ) @ ( top_top @ ( set @ B ) ) ) )
=> ( has_derivative @ A @ C
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ^ [X2: A] : ( G4 @ ( F7 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_compose
thf(fact_7330_has__derivative__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [I6: set @ A,F2: A > B > C,F7: A > B > C,F4: filter @ B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( has_derivative @ B @ C @ ( F2 @ I2 ) @ ( F7 @ I2 ) @ F4 ) )
=> ( has_derivative @ B @ C
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X2 )
@ I6 )
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F7 @ I4 @ X2 )
@ I6 )
@ F4 ) ) ) ).
% has_derivative_sum
thf(fact_7331_has__derivative__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,F4: filter @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) )
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F7 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_minus
thf(fact_7332_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C2: B,F4: filter @ A] :
( has_derivative @ A @ B
@ ^ [X2: A] : C2
@ ^ [X2: A] : ( zero_zero @ B )
@ F4 ) ) ).
% has_derivative_const
thf(fact_7333_has__derivative__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F4: filter @ A] :
( has_derivative @ A @ A
@ ^ [X2: A] : X2
@ ^ [X2: A] : X2
@ F4 ) ) ).
% has_derivative_ident
thf(fact_7334_has__derivative__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [G: C > real,G4: C > real,F4: filter @ C] :
( ( has_derivative @ C @ real @ G @ G4 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( real_Vector_of_real @ A @ ( G @ X2 ) )
@ ^ [X2: C] : ( real_Vector_of_real @ A @ ( G4 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_of_real
thf(fact_7335_has__derivative__scaleR__right,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > B,G4: C > B,F4: filter @ C,R3: real] :
( ( has_derivative @ C @ B @ G @ G4 @ F4 )
=> ( has_derivative @ C @ B
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ R3 @ ( G @ X2 ) )
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ R3 @ ( G4 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_scaleR_right
thf(fact_7336_has__derivative__scaleR__left,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > real,G4: C > real,F4: filter @ C,X: B] :
( ( has_derivative @ C @ real @ G @ G4 @ F4 )
=> ( has_derivative @ C @ B
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ ( G @ X2 ) @ X )
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ ( G4 @ X2 ) @ X )
@ F4 ) ) ) ).
% has_derivative_scaleR_left
thf(fact_7337_has__derivative__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,F4: filter @ A,G: A > B,G4: A > B] :
( ( has_derivative @ A @ B @ F2 @ F7 @ F4 )
=> ( ( has_derivative @ A @ B @ G @ G4 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [X2: A] : ( minus_minus @ B @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ F4 ) ) ) ) ).
% has_derivative_diff
thf(fact_7338_has__derivative__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,F4: filter @ A,G: A > B,G4: A > B] :
( ( has_derivative @ A @ B @ F2 @ F7 @ F4 )
=> ( ( has_derivative @ A @ B @ G @ G4 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [X2: A] : ( plus_plus @ B @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ F4 ) ) ) ) ).
% has_derivative_add
thf(fact_7339_has__derivative__mult__right,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,G4: C > A,F4: filter @ C,X: A] :
( ( has_derivative @ C @ A @ G @ G4 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( times_times @ A @ X @ ( G @ X2 ) )
@ ^ [X2: C] : ( times_times @ A @ X @ ( G4 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_mult_right
thf(fact_7340_has__derivative__mult__left,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,G4: C > A,F4: filter @ C,Y: A] :
( ( has_derivative @ C @ A @ G @ G4 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( times_times @ A @ ( G @ X2 ) @ Y )
@ ^ [X2: C] : ( times_times @ A @ ( G4 @ X2 ) @ Y )
@ F4 ) ) ) ).
% has_derivative_mult_left
thf(fact_7341_has__derivative__in__compose2,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [T2: set @ A,G: A > B,G4: A > A > B,F2: C > A,S: set @ C,X: C,F7: C > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ T2 )
=> ( has_derivative @ A @ B @ G @ ( G4 @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ S ) @ T2 )
=> ( ( member @ C @ X @ S )
=> ( ( has_derivative @ C @ A @ F2 @ F7 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( has_derivative @ C @ B
@ ^ [X2: C] : ( G @ ( F2 @ X2 ) )
@ ^ [Y4: C] : ( G4 @ ( F2 @ X ) @ ( F7 @ Y4 ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ) ) ).
% has_derivative_in_compose2
thf(fact_7342_has__derivative__exp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( exp @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( exp @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_exp
thf(fact_7343_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D4 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D4 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_7344_has__derivative__sin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( sin @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( cos @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_sin
thf(fact_7345_has__derivative__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,Db: A,X: A,S: set @ A] :
( ( has_derivative @ A @ A @ G @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ A
@ ^ [X2: A] : ( sinh @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( times_times @ A @ ( cosh @ A @ ( G @ X ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_sinh
thf(fact_7346_has__derivative__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,Db: A,X: A,S: set @ A] :
( ( has_derivative @ A @ A @ G @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ A
@ ^ [X2: A] : ( cosh @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( times_times @ A @ ( sinh @ A @ ( G @ X ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_cosh
thf(fact_7347_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D4 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D4 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_7348_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D4 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D4 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_7349_ivl__disj__un__two_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U2 ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_7350_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_7351_ivl__disj__un__one_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U2 ) )
= ( set_ord_lessThan @ A @ U2 ) ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_7352_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,F7: C > A,X: C,S2: set @ C,G: C > A,G4: C > A] :
( ( has_derivative @ C @ A @ F2 @ F7 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( ( has_derivative @ C @ A @ G @ G4 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [H: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F7 @ H ) @ ( G @ X ) ) @ ( times_times @ A @ ( F2 @ X ) @ ( G4 @ H ) ) ) @ ( times_times @ A @ ( G @ X ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_7353_has__derivative__inverse_H,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,S2: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A @ ( inverse_inverse @ A )
@ ^ [H: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X ) @ H ) @ ( inverse_inverse @ A @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_inverse'
thf(fact_7354_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,X: C,F7: C > A,S2: set @ C] :
( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F2 @ F7 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ ^ [H: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X ) ) @ ( F7 @ H ) ) @ ( inverse_inverse @ A @ ( F2 @ X ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_7355_DERIV__compose__FDERIV,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: real > real,F7: real,G: A > real,X: A,G4: A > real,S: set @ A] :
( ( has_field_derivative @ real @ F2 @ F7 @ ( topolo174197925503356063within @ real @ ( G @ X ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( F2 @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ F7 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_compose_FDERIV
thf(fact_7356_has__derivative__cos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( cos @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( uminus_uminus @ real @ ( sin @ real @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_cos
thf(fact_7357_DERIV__isconst3,axiom,
! [A2: real,B2: real,X: real,Y: real,F2: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( member @ real @ Y @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( F2 @ X )
= ( F2 @ Y ) ) ) ) ) ) ).
% DERIV_isconst3
thf(fact_7358_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U2 ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_7359_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U2 ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U2 ) ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_7360_has__derivative__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,F7: A > B,X: A,S2: set @ A,N: nat] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ ^ [Y4: A] : ( times_times @ B @ ( times_times @ B @ ( semiring_1_of_nat @ B @ N ) @ ( F7 @ Y4 ) ) @ ( power_power @ B @ ( F2 @ X ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_power
thf(fact_7361_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G4: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( inverse_inverse @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_ln
thf(fact_7362_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,F7: C > A,X: C,S2: set @ C,G: C > A,G4: C > A] :
( ( has_derivative @ C @ A @ F2 @ F7 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( ( has_derivative @ C @ A @ G @ G4 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [H: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F2 @ X ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G @ X ) ) @ ( G4 @ H ) ) @ ( inverse_inverse @ A @ ( G @ X ) ) ) ) @ ( divide_divide @ A @ ( F7 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_7363_has__derivative__prod,axiom,
! [B: $tType,I5: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I6: set @ I5,F2: I5 > A > B,F7: I5 > A > B,X: A,S2: set @ A] :
( ! [I2: I5] :
( ( member @ I5 @ I2 @ I6 )
=> ( has_derivative @ A @ B @ ( F2 @ I2 ) @ ( F7 @ I2 ) @ ( topolo174197925503356063within @ A @ X @ S2 ) ) )
=> ( has_derivative @ A @ B
@ ^ [X2: A] :
( groups7121269368397514597t_prod @ I5 @ B
@ ^ [I4: I5] : ( F2 @ I4 @ X2 )
@ I6 )
@ ^ [Y4: A] :
( groups7311177749621191930dd_sum @ I5 @ B
@ ^ [I4: I5] :
( times_times @ B @ ( F7 @ I4 @ Y4 )
@ ( groups7121269368397514597t_prod @ I5 @ B
@ ^ [J3: I5] : ( F2 @ J3 @ X )
@ ( minus_minus @ ( set @ I5 ) @ I6 @ ( insert @ I5 @ I4 @ ( bot_bot @ ( set @ I5 ) ) ) ) ) )
@ I6 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_prod
thf(fact_7364_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X: A,X5: set @ A,F2: A > real,F7: A > real] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ X5 ) )
=> ( ( has_derivative @ A @ real @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ X5 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( member @ A @ X @ X5 )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( G @ X2 ) @ ( F2 @ X2 ) )
@ ^ [H: A] : ( times_times @ real @ ( powr @ real @ ( G @ X ) @ ( F2 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F7 @ H ) @ ( ln_ln @ real @ ( G @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G4 @ H ) @ ( F2 @ X ) ) @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ X5 ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_7365_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G4: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( sqrt @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_7366_has__derivative__arctan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G4: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arctan @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_arctan
thf(fact_7367_has__derivative__tan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G4: A > real,S: set @ A] :
( ( ( cos @ real @ ( G @ X ) )
!= ( zero_zero @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( tan @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( inverse_inverse @ real @ ( power_power @ real @ ( cos @ real @ ( G @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_tan
thf(fact_7368_DERIV__series_H,axiom,
! [F2: real > nat > real,F7: real > nat > real,X0: real,A2: real,B2: real,L6: nat > real] :
( ! [N2: nat] :
( has_field_derivative @ real
@ ^ [X2: real] : ( F2 @ X2 @ N2 )
@ ( F7 @ X0 @ N2 )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( summable @ real @ ( F2 @ X3 ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( summable @ real @ ( F7 @ X0 ) )
=> ( ( summable @ real @ L6 )
=> ( ! [N2: nat,X3: real,Y3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( member @ real @ Y3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( F2 @ X3 @ N2 ) @ ( F2 @ Y3 @ N2 ) ) ) @ ( times_times @ real @ ( L6 @ N2 ) @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y3 ) ) ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( suminf @ real @ ( F2 @ X2 ) )
@ ( suminf @ real @ ( F7 @ X0 ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_7369_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G4: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X ) )
=> ( ( ord_less @ real @ ( G @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arccos @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_7370_has__derivative__floor,axiom,
! [Aa: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( archim2362893244070406136eiling @ Aa )
& ( topolo2564578578187576103pology @ Aa ) )
=> ! [G: A > real,X: A,F2: real > Aa,G4: A > real,S: set @ A] :
( ( topolo3448309680560233919inuous @ real @ Aa @ ( topolo174197925503356063within @ real @ ( G @ X ) @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ~ ( member @ Aa @ ( F2 @ ( G @ X ) ) @ ( ring_1_Ints @ Aa ) )
=> ( ( has_derivative @ A @ real @ G @ G4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ Aa @ ( F2 @ ( G @ X2 ) ) ) )
@ ^ [X2: A] : ( times_times @ real @ ( G4 @ X2 ) @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_floor
thf(fact_7371_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( filterlim @ A @ A
@ ^ [H: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X @ H ) @ N3 ) @ ( power_power @ A @ X @ N3 ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_7372_tendsto__const,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [K: A,F4: filter @ B] :
( filterlim @ B @ A
@ ^ [X2: B] : K
@ ( topolo7230453075368039082e_nhds @ A @ K )
@ F4 ) ) ).
% tendsto_const
thf(fact_7373_tendsto__ident__at,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A,S: set @ A] :
( filterlim @ A @ A
@ ^ [X2: A] : X2
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( topolo174197925503356063within @ A @ A2 @ S ) ) ) ).
% tendsto_ident_at
thf(fact_7374_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L: A,F4: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) )
@ F4 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_7375_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L: A,F4: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C2 ) )
@ F4 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_7376_power__tendsto__0__iff,axiom,
! [A: $tType,N: nat,F2: A > real,F4: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real
@ ^ [X2: A] : ( power_power @ real @ ( F2 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% power_tendsto_0_iff
thf(fact_7377_continuous__within__compose2,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topological_t2_space @ B ) )
=> ! [X: A,S: set @ A,F2: A > B,G: B > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ ( F2 @ X ) @ ( image @ A @ B @ F2 @ S ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_within_compose2
thf(fact_7378_tendsto__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: A > B,L: filter @ B,X: A,S2: set @ A,T5: set @ A] :
( ( filterlim @ A @ B @ F2 @ L @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T5 @ S2 )
=> ( filterlim @ A @ B @ F2 @ L @ ( topolo174197925503356063within @ A @ X @ T5 ) ) ) ) ) ).
% tendsto_within_subset
thf(fact_7379_continuous__ident,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,S2: set @ A] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S2 )
@ ^ [X2: A] : X2 ) ) ).
% continuous_ident
thf(fact_7380_filterlim__at__within__If,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: A > B,G6: filter @ B,X: A,A3: set @ A,P: A > $o,G: A > B] :
( ( filterlim @ A @ B @ F2 @ G6 @ ( topolo174197925503356063within @ A @ X @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) ) )
=> ( ( filterlim @ A @ B @ G @ G6
@ ( topolo174197925503356063within @ A @ X
@ ( inf_inf @ ( set @ A ) @ A3
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( if @ B @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ G6
@ ( topolo174197925503356063within @ A @ X @ A3 ) ) ) ) ) ).
% filterlim_at_within_If
thf(fact_7381_has__field__derivativeD,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S2: set @ A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( filterlim @ A @ A
@ ^ [Y4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ Y4 ) @ ( F2 @ X ) ) @ ( minus_minus @ A @ Y4 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_field_derivativeD
thf(fact_7382_has__field__derivative__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A,S2: set @ A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ S2 ) )
= ( filterlim @ A @ A
@ ^ [Y4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ Y4 ) @ ( F2 @ X ) ) @ ( minus_minus @ A @ Y4 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_field_derivative_iff
thf(fact_7383_isCont__snd,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F2: A > ( product_prod @ B @ C )] :
( ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( product_snd @ B @ C @ ( F2 @ X2 ) ) ) ) ) ).
% isCont_snd
thf(fact_7384_isCont__fst,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F2: A > ( product_prod @ B @ C )] :
( ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( product_fst @ B @ C @ ( F2 @ X2 ) ) ) ) ) ).
% isCont_fst
thf(fact_7385_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > A,A2: A,D: A] :
( ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ A2 @ H ) ) @ ( F2 @ A2 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ X2 ) @ ( F2 @ A2 ) ) @ ( minus_minus @ A @ X2 @ A2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_7386_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F2 )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ X @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_7387_LIM__isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_isCont_iff
thf(fact_7388_LIM__offset__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L6: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero
thf(fact_7389_LIM__offset__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A2: A,L6: B] :
( ( filterlim @ A @ B
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero_cancel
thf(fact_7390_isCont__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F2: A > B,G: B > C,L: C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ L ) @ ( topolo174197925503356063within @ B @ ( F2 @ A2 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ D2 ) )
=> ( ( F2 @ X3 )
!= ( F2 @ A2 ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_7391_LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,B2: B,A2: A,G: B > C,C2: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ D2 ) )
=> ( ( F2 @ X3 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose2
thf(fact_7392_LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L6: B,A2: A,R3: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [S3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S3 )
& ! [X6: A] :
( ( ( X6 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X6 @ A2 ) ) @ S3 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X6 ) @ L6 ) ) @ R3 ) ) ) ) ) ) ).
% LIM_D
thf(fact_7393_LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B,L6: B] :
( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [S10: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S10 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ S10 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X3 ) @ L6 ) ) @ R ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_I
thf(fact_7394_LIM__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L6: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S8 )
& ! [X2: A] :
( ( ( X2 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ A2 ) ) @ S8 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X2 ) @ L6 ) ) @ R5 ) ) ) ) ) ) ) ).
% LIM_eq
thf(fact_7395_LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [R2: real,A2: A,F2: A > B,G: A > B,L: B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ R2 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) ) )
=> ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_equal2
thf(fact_7396_real__LIM__sandwich__zero,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: A > real,A2: A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% real_LIM_sandwich_zero
thf(fact_7397_LIM__not__zero,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( zero @ Aa )
& ( topological_t2_space @ Aa ) )
=> ! [K: Aa,A2: A] :
( ( K
!= ( zero_zero @ Aa ) )
=> ~ ( filterlim @ A @ Aa
@ ^ [X2: A] : K
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_7398_isCont__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [A2: C,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( real_Vector_of_real @ A @ ( G @ X2 ) ) ) ) ) ).
% isCont_of_real
thf(fact_7399_filterlim__at__If,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: A > B,G6: filter @ B,X: A,P: A > $o,G: A > B] :
( ( filterlim @ A @ B @ F2 @ G6 @ ( topolo174197925503356063within @ A @ X @ ( collect @ A @ P ) ) )
=> ( ( filterlim @ A @ B @ G @ G6
@ ( topolo174197925503356063within @ A @ X
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( if @ B @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ G6
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% filterlim_at_If
thf(fact_7400_isCont__o2,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topological_t2_space @ B ) )
=> ! [A2: A,F2: A > B,G: B > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ ( F2 @ A2 ) @ ( top_top @ ( set @ B ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_o2
thf(fact_7401_LIM__const__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo8386298272705272623_space @ A ) )
=> ! [K: B,L6: B,A2: A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : K
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( K = L6 ) ) ) ).
% LIM_const_eq
thf(fact_7402_tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G: A > B,L: A,F2: C > A,F4: filter @ C] :
( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( G @ L ) ) @ ( topolo174197925503356063within @ A @ L @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( G @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( G @ L ) )
@ F4 ) ) ) ) ).
% tendsto_compose
thf(fact_7403_LIM__const__not__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( topological_t2_space @ B ) )
=> ! [K: B,L6: B,A2: A] :
( ( K != L6 )
=> ~ ( filterlim @ A @ B
@ ^ [X2: A] : K
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_const_not_eq
thf(fact_7404_isCont__tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topological_t2_space @ A ) )
=> ! [L: A,G: A > B,F2: C > A,F4: filter @ C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ L @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( G @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( G @ L ) )
@ F4 ) ) ) ) ).
% isCont_tendsto_compose
thf(fact_7405_continuous__within__compose3,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( topolo4958980785337419405_space @ B )
& ( topological_t2_space @ A ) )
=> ! [F2: C > A,X: C,G: A > B,S: set @ C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( F2 @ X ) @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ S ) @ F2 )
=> ( topolo3448309680560233919inuous @ C @ B @ ( topolo174197925503356063within @ C @ X @ S )
@ ^ [X2: C] : ( G @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_within_compose3
thf(fact_7406_isCont__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) ) ) ) ).
% isCont_norm
thf(fact_7407_isCont__scaleR,axiom,
! [C: $tType,D3: $tType] :
( ( ( topological_t2_space @ D3 )
& ( real_V822414075346904944vector @ C ) )
=> ! [A2: D3,F2: D3 > real,G: D3 > C] :
( ( topolo3448309680560233919inuous @ D3 @ real @ ( topolo174197925503356063within @ D3 @ A2 @ ( top_top @ ( set @ D3 ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ D3 @ C @ ( topolo174197925503356063within @ D3 @ A2 @ ( top_top @ ( set @ D3 ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ D3 @ C @ ( topolo174197925503356063within @ D3 @ A2 @ ( top_top @ ( set @ D3 ) ) )
@ ^ [X2: D3] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% isCont_scaleR
thf(fact_7408_IVT,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,A2: A,Y: B,B2: A] :
( ( ord_less_eq @ B @ ( F2 @ A2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F2 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT
thf(fact_7409_IVT2,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F2: A > B,B2: A,Y: B,A2: A] :
( ( ord_less_eq @ B @ ( F2 @ B2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F2 @ A2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F2 ) )
=> ? [X3: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 )
& ( ( F2 @ X3 )
= Y ) ) ) ) ) ) ) ).
% IVT2
thf(fact_7410_isCont__Lb__Ub,axiom,
! [A2: real,B2: real,F2: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [L7: real,M8: real] :
( ! [X6: real] :
( ( ( ord_less_eq @ real @ A2 @ X6 )
& ( ord_less_eq @ real @ X6 @ B2 ) )
=> ( ( ord_less_eq @ real @ L7 @ ( F2 @ X6 ) )
& ( ord_less_eq @ real @ ( F2 @ X6 ) @ M8 ) ) )
& ! [Y2: real] :
( ( ( ord_less_eq @ real @ L7 @ Y2 )
& ( ord_less_eq @ real @ Y2 @ M8 ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ( F2 @ X3 )
= Y2 ) ) ) ) ) ) ).
% isCont_Lb_Ub
thf(fact_7411_LIM__imp__LIM,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L: B,A2: A,G: A > C,M: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( minus_minus @ C @ ( G @ X3 ) @ M ) ) @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X3 ) @ L ) ) ) )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ M ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_imp_LIM
thf(fact_7412_LIM__offset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L6: B,A2: A,K: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( F2 @ ( plus_plus @ A @ X2 @ K ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ K ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset
thf(fact_7413_LIM__fun__gt__zero,axiom,
! [F2: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
& ! [X6: real] :
( ( ( X6 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X6 ) ) @ R ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X6 ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_7414_LIM__fun__not__zero,axiom,
! [F2: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( L
!= ( zero_zero @ real ) )
=> ? [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
& ! [X6: real] :
( ( ( X6 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X6 ) ) @ R ) )
=> ( ( F2 @ X6 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_7415_LIM__fun__less__zero,axiom,
! [F2: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
& ! [X6: real] :
( ( ( X6 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X6 ) ) @ R ) )
=> ( ord_less @ real @ ( F2 @ X6 ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_7416_isCont__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F2: A > B,G: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% isCont_Pair
thf(fact_7417_tendsto__artanh,axiom,
! [A: $tType,F2: A > real,A2: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ A2 )
=> ( ( ord_less @ real @ A2 @ ( one_one @ real ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( artanh @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( artanh @ real @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_artanh
thf(fact_7418_tendsto__log,axiom,
! [A: $tType,F2: A > real,A2: real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( log @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( log @ A2 @ B2 ) )
@ F4 ) ) ) ) ) ) ).
% tendsto_log
thf(fact_7419_tendsto__arcosh,axiom,
! [B: $tType,F2: B > real,A2: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arcosh @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F4 ) ) ) ).
% tendsto_arcosh
thf(fact_7420_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F2: A > B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_null_power
thf(fact_7421_has__derivative__Re,axiom,
! [C: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [G: C > complex,G4: C > complex,F4: filter @ C] :
( ( has_derivative @ C @ complex @ G @ G4 @ F4 )
=> ( has_derivative @ C @ real
@ ^ [X2: C] : ( re @ ( G @ X2 ) )
@ ^ [X2: C] : ( re @ ( G4 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_Re
thf(fact_7422_has__derivative__Im,axiom,
! [C: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [G: C > complex,G4: C > complex,F4: filter @ C] :
( ( has_derivative @ C @ complex @ G @ G4 @ F4 )
=> ( has_derivative @ C @ real
@ ^ [X2: C] : ( im @ ( G @ X2 ) )
@ ^ [X2: C] : ( im @ ( G4 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_Im
thf(fact_7423_has__derivative__cnj,axiom,
! [C: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [G: C > complex,G4: C > complex,F4: filter @ C] :
( ( has_derivative @ C @ complex @ G @ G4 @ F4 )
=> ( has_derivative @ C @ complex
@ ^ [X2: C] : ( cnj @ ( G @ X2 ) )
@ ^ [X2: C] : ( cnj @ ( G4 @ X2 ) )
@ F4 ) ) ) ).
% has_derivative_cnj
thf(fact_7424_filterlim__INF_H,axiom,
! [C: $tType,B: $tType,A: $tType,X: A,A3: set @ A,F2: B > C,F4: filter @ C,G6: A > ( filter @ B )] :
( ( member @ A @ X @ A3 )
=> ( ( filterlim @ B @ C @ F2 @ F4 @ ( G6 @ X ) )
=> ( filterlim @ B @ C @ F2 @ F4 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ G6 @ A3 ) ) ) ) ) ).
% filterlim_INF'
thf(fact_7425_filterlim__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F2: A > B,G6: C > ( filter @ B ),B3: set @ C,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ C @ ( filter @ B ) @ G6 @ B3 ) ) @ F4 )
= ( ! [X2: C] :
( ( member @ C @ X2 @ B3 )
=> ( filterlim @ A @ B @ F2 @ ( G6 @ X2 ) @ F4 ) ) ) ) ).
% filterlim_INF
thf(fact_7426_isCont__real__sqrt,axiom,
! [X: real] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ sqrt ) ).
% isCont_real_sqrt
thf(fact_7427_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G: A > B,F33: filter @ B,F24: filter @ A,F2: C > A,F13: filter @ C] :
( ( filterlim @ A @ B @ G @ F33 @ F24 )
=> ( ( filterlim @ C @ A @ F2 @ F24 @ F13 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( G @ ( F2 @ X2 ) )
@ F33
@ F13 ) ) ) ).
% filterlim_compose
thf(fact_7428_filterlim__ident,axiom,
! [A: $tType,F4: filter @ A] :
( filterlim @ A @ A
@ ^ [X2: A] : X2
@ F4
@ F4 ) ).
% filterlim_ident
thf(fact_7429_tendsto__const__iff,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ B,A2: A,B2: A] :
( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : A2
@ ( topolo7230453075368039082e_nhds @ A @ B2 )
@ F4 )
= ( A2 = B2 ) ) ) ) ).
% tendsto_const_iff
thf(fact_7430_continuous__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo4987421752381908075d_mult @ C ) )
=> ! [I6: set @ A,F4: filter @ B,F2: A > B > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( topolo3448309680560233919inuous @ B @ C @ F4 @ ( F2 @ I2 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F4
@ ^ [X2: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X2 )
@ I6 ) ) ) ) ).
% continuous_prod'
thf(fact_7431_continuous__prod,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S2: set @ A,F4: filter @ B,F2: A > B > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ S2 )
=> ( topolo3448309680560233919inuous @ B @ C @ F4 @ ( F2 @ I2 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F4
@ ^ [X2: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X2 )
@ S2 ) ) ) ) ).
% continuous_prod
thf(fact_7432_tendsto__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( topolo4987421752381908075d_mult @ C )
=> ! [I6: set @ A,F2: A > B > C,A2: A > C,F4: filter @ B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( filterlim @ B @ C @ ( F2 @ I2 ) @ ( topolo7230453075368039082e_nhds @ C @ ( A2 @ I2 ) ) @ F4 ) )
=> ( filterlim @ B @ C
@ ^ [X2: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X2 )
@ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7121269368397514597t_prod @ A @ C @ A2 @ I6 ) )
@ F4 ) ) ) ).
% tendsto_prod'
thf(fact_7433_tendsto__prod,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S2: set @ A,F2: A > B > C,L6: A > C,F4: filter @ B] :
( ! [I2: A] :
( ( member @ A @ I2 @ S2 )
=> ( filterlim @ B @ C @ ( F2 @ I2 ) @ ( topolo7230453075368039082e_nhds @ C @ ( L6 @ I2 ) ) @ F4 ) )
=> ( filterlim @ B @ C
@ ^ [X2: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X2 )
@ S2 )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7121269368397514597t_prod @ A @ C @ L6 @ S2 ) )
@ F4 ) ) ) ).
% tendsto_prod
thf(fact_7434_tendsto__one__prod_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo4987421752381908075d_mult @ C )
=> ! [I6: set @ B,F2: A > B > C,F4: filter @ A] :
( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( F2 @ X2 @ I2 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F4 ) )
=> ( filterlim @ A @ C
@ ^ [I4: A] : ( groups7121269368397514597t_prod @ B @ C @ ( F2 @ I4 ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F4 ) ) ) ).
% tendsto_one_prod'
thf(fact_7435_continuous__max,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F4: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( ord_max @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_max
thf(fact_7436_tendsto__max,axiom,
! [A: $tType,B: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X5: B > A,X: A,Net: filter @ B,Y6: B > A,Y: A] :
( ( filterlim @ B @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ Net )
=> ( ( filterlim @ B @ A @ Y6 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ Net )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( ord_max @ A @ ( X5 @ X2 ) @ ( Y6 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( ord_max @ A @ X @ Y ) )
@ Net ) ) ) ) ).
% tendsto_max
thf(fact_7437_continuous__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F4: filter @ A,F2: A > B,G: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ F4 @ G )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F4
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_Pair
thf(fact_7438_tendsto__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,A2: B,F4: filter @ A,G: A > C,B2: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F4 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_Pair
thf(fact_7439_tendsto__rabs__zero,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( abs_abs @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ).
% tendsto_rabs_zero
thf(fact_7440_tendsto__rabs__zero__iff,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( abs_abs @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ).
% tendsto_rabs_zero_iff
thf(fact_7441_tendsto__rabs__zero__cancel,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( abs_abs @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ).
% tendsto_rabs_zero_cancel
thf(fact_7442_tendsto__rabs,axiom,
! [A: $tType,F2: A > real,L: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( abs_abs @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( abs_abs @ real @ L ) )
@ F4 ) ) ).
% tendsto_rabs
thf(fact_7443_tendsto__mult__one,axiom,
! [B: $tType,D3: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F2: D3 > B,F4: filter @ D3,G: D3 > B] :
( ( filterlim @ D3 @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F4 )
=> ( ( filterlim @ D3 @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F4 )
=> ( filterlim @ D3 @ B
@ ^ [X2: D3] : ( times_times @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) )
@ F4 ) ) ) ) ).
% tendsto_mult_one
thf(fact_7444_continuous__mult,axiom,
! [A: $tType,D3: $tType] :
( ( ( topological_t2_space @ D3 )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ D3,F2: D3 > A,G: D3 > A] :
( ( topolo3448309680560233919inuous @ D3 @ A @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D3 @ A @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D3 @ A @ F4
@ ^ [X2: D3] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_mult
thf(fact_7445_continuous__mult_H,axiom,
! [B: $tType,D3: $tType] :
( ( ( topological_t2_space @ D3 )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [F4: filter @ D3,F2: D3 > B,G: D3 > B] :
( ( topolo3448309680560233919inuous @ D3 @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D3 @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D3 @ B @ F4
@ ^ [X2: D3] : ( times_times @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_mult'
thf(fact_7446_continuous__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ B,F2: B > A,C2: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ B @ A @ F4
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_mult_left
thf(fact_7447_continuous__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ B,F2: B > A,C2: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ B @ A @ F4
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 ) ) ) ) ).
% continuous_mult_right
thf(fact_7448_tendsto__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,L: A,F4: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C2 ) )
@ F4 ) ) ) ).
% tendsto_mult_right
thf(fact_7449_tendsto__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,L: A,F4: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) )
@ F4 ) ) ) ).
% tendsto_mult_left
thf(fact_7450_tendsto__mult,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_mult
thf(fact_7451_continuous__add,axiom,
! [B: $tType,D3: $tType] :
( ( ( topological_t2_space @ D3 )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [F4: filter @ D3,F2: D3 > B,G: D3 > B] :
( ( topolo3448309680560233919inuous @ D3 @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D3 @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D3 @ B @ F4
@ ^ [X2: D3] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_add
thf(fact_7452_tendsto__add__const__iff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [C2: A,F2: B > A,D4: A,F4: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ C2 @ D4 ) )
@ F4 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ D4 ) @ F4 ) ) ) ).
% tendsto_add_const_iff
thf(fact_7453_tendsto__add,axiom,
! [A: $tType,B: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_add
thf(fact_7454_continuous__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [F4: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_diff
thf(fact_7455_tendsto__diff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( minus_minus @ A @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_diff
thf(fact_7456_lim__cnj,axiom,
! [A: $tType,F2: A > complex,L: complex,F4: filter @ A] :
( ( filterlim @ A @ complex
@ ^ [X2: A] : ( cnj @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ complex @ ( cnj @ L ) )
@ F4 )
= ( filterlim @ A @ complex @ F2 @ ( topolo7230453075368039082e_nhds @ complex @ L ) @ F4 ) ) ).
% lim_cnj
thf(fact_7457_tendsto__cnj,axiom,
! [C: $tType,G: C > complex,A2: complex,F4: filter @ C] :
( ( filterlim @ C @ complex @ G @ ( topolo7230453075368039082e_nhds @ complex @ A2 ) @ F4 )
=> ( filterlim @ C @ complex
@ ^ [X2: C] : ( cnj @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ complex @ ( cnj @ A2 ) )
@ F4 ) ) ).
% tendsto_cnj
thf(fact_7458_tendsto__complex__iff,axiom,
! [A: $tType,F2: A > complex,X: complex,F4: filter @ A] :
( ( filterlim @ A @ complex @ F2 @ ( topolo7230453075368039082e_nhds @ complex @ X ) @ F4 )
= ( ( filterlim @ A @ real
@ ^ [X2: A] : ( re @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( re @ X ) )
@ F4 )
& ( filterlim @ A @ real
@ ^ [X2: A] : ( im @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( im @ X ) )
@ F4 ) ) ) ).
% tendsto_complex_iff
thf(fact_7459_tendsto__Im,axiom,
! [C: $tType,G: C > complex,A2: complex,F4: filter @ C] :
( ( filterlim @ C @ complex @ G @ ( topolo7230453075368039082e_nhds @ complex @ A2 ) @ F4 )
=> ( filterlim @ C @ real
@ ^ [X2: C] : ( im @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( im @ A2 ) )
@ F4 ) ) ).
% tendsto_Im
thf(fact_7460_tendsto__real__root,axiom,
! [A: $tType,F2: A > real,X: real,F4: filter @ A,N: nat] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( root @ N @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( root @ N @ X ) )
@ F4 ) ) ).
% tendsto_real_root
thf(fact_7461_continuous__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X2: C] : ( sinh @ A @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_sinh
thf(fact_7462_tendsto__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A2: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( sinh @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sinh @ A @ A2 ) )
@ F4 ) ) ) ).
% tendsto_sinh
thf(fact_7463_tendsto__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A2: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( cosh @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cosh @ A @ A2 ) )
@ F4 ) ) ) ).
% tendsto_cosh
thf(fact_7464_tendsto__scaleR,axiom,
! [C: $tType,D3: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [F2: D3 > real,A2: real,F4: filter @ D3,G: D3 > C,B2: C] :
( ( filterlim @ D3 @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ D3 @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F4 )
=> ( filterlim @ D3 @ C
@ ^ [X2: D3] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( real_V8093663219630862766scaleR @ C @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_scaleR
thf(fact_7465_continuous__scaleR,axiom,
! [C: $tType,D3: $tType] :
( ( ( topological_t2_space @ D3 )
& ( real_V822414075346904944vector @ C ) )
=> ! [F4: filter @ D3,F2: D3 > real,G: D3 > C] :
( ( topolo3448309680560233919inuous @ D3 @ real @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D3 @ C @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D3 @ C @ F4
@ ^ [X2: D3] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_scaleR
thf(fact_7466_tendsto__power__strong,axiom,
! [B: $tType,C: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F2: C > B,A2: B,F4: filter @ C,G: C > nat,B2: nat] :
( ( filterlim @ C @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( ( filterlim @ C @ nat @ G @ ( topolo7230453075368039082e_nhds @ nat @ B2 ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( power_power @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_power_strong
thf(fact_7467_continuous__power_H,axiom,
! [B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( topolo1898628316856586783d_mult @ B ) )
=> ! [F4: filter @ C,F2: C > B,G: C > nat] :
( ( topolo3448309680560233919inuous @ C @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ nat @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ B @ F4
@ ^ [X2: C] : ( power_power @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_power'
thf(fact_7468_tendsto__power,axiom,
! [B: $tType,A: $tType] :
( ( ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F2: A > B,A2: B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A2 @ N ) )
@ F4 ) ) ) ).
% tendsto_power
thf(fact_7469_continuous__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F4: filter @ A,F2: A > B,N: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N ) ) ) ) ).
% continuous_power
thf(fact_7470_tendsto__arsinh,axiom,
! [B: $tType,F2: B > real,A2: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arsinh @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arsinh @ real @ A2 ) )
@ F4 ) ) ).
% tendsto_arsinh
thf(fact_7471_tendsto__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,A2: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( sin @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( sin @ B @ A2 ) )
@ F4 ) ) ) ).
% tendsto_sin
thf(fact_7472_tendsto__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,A2: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( cos @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( cos @ B @ A2 ) )
@ F4 ) ) ) ).
% tendsto_cos
thf(fact_7473_continuous__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_norm
thf(fact_7474_tendsto__norm,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,A2: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( real_V7770717601297561774m_norm @ B @ A2 ) )
@ F4 ) ) ) ).
% tendsto_norm
thf(fact_7475_tendsto__Complex,axiom,
! [A: $tType,F2: A > real,A2: real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( filterlim @ A @ complex
@ ^ [X2: A] : ( complex2 @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ complex @ ( complex2 @ A2 @ B2 ) )
@ F4 ) ) ) ).
% tendsto_Complex
thf(fact_7476_continuous__const,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F4: filter @ A,C2: B] :
( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : C2 ) ) ).
% continuous_const
thf(fact_7477_tendsto__of__real__iff,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > real,C2: real,F4: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( real_Vector_of_real @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( real_Vector_of_real @ A @ C2 ) )
@ F4 )
= ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 ) ) ) ).
% tendsto_of_real_iff
thf(fact_7478_continuous__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F4: filter @ C,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X2: C] : ( real_Vector_of_real @ A @ ( G @ X2 ) ) ) ) ) ).
% continuous_of_real
thf(fact_7479_tendsto__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [G: C > real,A2: real,F4: filter @ C] :
( ( filterlim @ C @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( real_Vector_of_real @ A @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( real_Vector_of_real @ A @ A2 ) )
@ F4 ) ) ) ).
% tendsto_of_real
thf(fact_7480_tendsto__arctan,axiom,
! [A: $tType,F2: A > real,X: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( arctan @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arctan @ X ) )
@ F4 ) ) ).
% tendsto_arctan
thf(fact_7481_tendsto__real__sqrt,axiom,
! [A: $tType,F2: A > real,X: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( sqrt @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( sqrt @ X ) )
@ F4 ) ) ).
% tendsto_real_sqrt
thf(fact_7482_tendsto__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A2: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( exp @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( exp @ A @ A2 ) )
@ F4 ) ) ) ).
% tendsto_exp
thf(fact_7483_continuous__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X2: C] : ( exp @ A @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_exp
thf(fact_7484_continuous__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( sin @ B @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_sin
thf(fact_7485_continuous__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( cos @ B @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_cos
thf(fact_7486_continuous__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X2: C] : ( cosh @ A @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_cosh
thf(fact_7487_tendsto__Re,axiom,
! [C: $tType,G: C > complex,A2: complex,F4: filter @ C] :
( ( filterlim @ C @ complex @ G @ ( topolo7230453075368039082e_nhds @ complex @ A2 ) @ F4 )
=> ( filterlim @ C @ real
@ ^ [X2: C] : ( re @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( re @ A2 ) )
@ F4 ) ) ).
% tendsto_Re
thf(fact_7488_tendsto__of__int__ceiling,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( ring_1 @ C )
& ( topolo4958980785337419405_space @ C )
& ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( ring_1_of_int @ C @ ( archimedean_ceiling @ B @ ( F2 @ X2 ) ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( ring_1_of_int @ C @ ( archimedean_ceiling @ B @ L ) ) )
@ F4 ) ) ) ) ).
% tendsto_of_int_ceiling
thf(fact_7489_tendsto__of__int__floor,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( ring_1 @ C )
& ( topolo4958980785337419405_space @ C )
& ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( ring_1_of_int @ C @ ( archim6421214686448440834_floor @ B @ ( F2 @ X2 ) ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( ring_1_of_int @ C @ ( archim6421214686448440834_floor @ B @ L ) ) )
@ F4 ) ) ) ) ).
% tendsto_of_int_floor
thf(fact_7490_tendsto__divide,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A2 @ B2 ) )
@ F4 ) ) ) ) ) ).
% tendsto_divide
thf(fact_7491_tendsto__divide__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,F4: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_divide_zero
thf(fact_7492_tendsto__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_inverse
thf(fact_7493_tendsto__sgn,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( L
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( sgn_sgn @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) )
@ F4 ) ) ) ) ).
% tendsto_sgn
thf(fact_7494_tendsto__powr,axiom,
! [A: $tType,F2: A > real,A2: real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( A2
!= ( zero_zero @ real ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_powr
thf(fact_7495_tendsto__ln,axiom,
! [A: $tType,F2: A > real,A2: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( A2
!= ( zero_zero @ real ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( ln_ln @ real @ A2 ) )
@ F4 ) ) ) ).
% tendsto_ln
thf(fact_7496_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_7497_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_7498_tendsto__norm__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_norm_zero
thf(fact_7499_tendsto__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: A > A,A2: A,F4: filter @ A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( ( cos @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_tan
thf(fact_7500_tendsto__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A2: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( ( cosh @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( tanh @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_tanh
thf(fact_7501_tendsto__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: A > A,A2: A,F4: filter @ A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( ( sin @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_cot
thf(fact_7502_tendsto__minus,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A2 ) )
@ F4 ) ) ) ).
% tendsto_minus
thf(fact_7503_tendsto__minus__cancel,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A2 ) )
@ F4 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 ) ) ) ).
% tendsto_minus_cancel
thf(fact_7504_tendsto__minus__cancel__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo1633459387980952147up_add @ B )
=> ! [F2: A > B,Y: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( uminus_uminus @ B @ Y ) ) @ F4 )
= ( filterlim @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y )
@ F4 ) ) ) ).
% tendsto_minus_cancel_left
thf(fact_7505_continuous__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [F4: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_minus
thf(fact_7506_Lim__transform__eq,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,G: B > A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
= ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 ) ) ) ) ).
% Lim_transform_eq
thf(fact_7507_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ).
% LIM_zero_cancel
thf(fact_7508_Lim__transform2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 ) ) ) ) ).
% Lim_transform2
thf(fact_7509_Lim__transform,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: B > A,A2: A,F4: filter @ B,F2: B > A] :
( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 ) ) ) ) ).
% Lim_transform
thf(fact_7510_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ).
% LIM_zero_iff
thf(fact_7511_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ).
% LIM_zero
thf(fact_7512_tendsto__add__zero,axiom,
! [B: $tType,D3: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F2: D3 > B,F4: filter @ D3,G: D3 > B] :
( ( filterlim @ D3 @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( filterlim @ D3 @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ D3 @ B
@ ^ [X2: D3] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_add_zero
thf(fact_7513_tendsto__mult__right__zero,axiom,
! [A: $tType,D3: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D3 > A,F4: filter @ D3,C2: A] :
( ( filterlim @ D3 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D3 @ A
@ ^ [X2: D3] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_7514_tendsto__mult__left__zero,axiom,
! [A: $tType,D3: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D3 > A,F4: filter @ D3,C2: A] :
( ( filterlim @ D3 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D3 @ A
@ ^ [X2: D3] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_7515_tendsto__mult__zero,axiom,
! [A: $tType,D3: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D3 > A,F4: filter @ D3,G: D3 > A] :
( ( filterlim @ D3 @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( ( filterlim @ D3 @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D3 @ A
@ ^ [X2: D3] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_7516_tendsto__fst,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > ( product_prod @ B @ C ),A2: product_prod @ B @ C,F4: filter @ A] :
( ( filterlim @ A @ ( product_prod @ B @ C ) @ F2 @ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ A2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( product_fst @ B @ C @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( product_fst @ B @ C @ A2 ) )
@ F4 ) ) ) ).
% tendsto_fst
thf(fact_7517_tendsto__snd,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > ( product_prod @ B @ C ),A2: product_prod @ B @ C,F4: filter @ A] :
( ( filterlim @ A @ ( product_prod @ B @ C ) @ F2 @ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ A2 ) @ F4 )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( product_snd @ B @ C @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( product_snd @ B @ C @ A2 ) )
@ F4 ) ) ) ).
% tendsto_snd
thf(fact_7518_continuous__fst,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F4: filter @ A,F2: A > ( product_prod @ B @ C )] :
( ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( product_fst @ B @ C @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_fst
thf(fact_7519_continuous__snd,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F4: filter @ A,F2: A > ( product_prod @ B @ C )] :
( ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ C @ F4
@ ^ [X2: A] : ( product_snd @ B @ C @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_snd
thf(fact_7520_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ B,F2: A > B > C,F4: filter @ A] :
( ! [I2: B] :
( ( member @ B @ I2 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( F2 @ X2 @ I2 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) )
=> ( filterlim @ A @ C
@ ^ [I4: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F2 @ I4 ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) ) ) ).
% tendsto_null_sum
thf(fact_7521_continuous__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [I6: set @ A,F4: filter @ B,F2: A > B > C] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( topolo3448309680560233919inuous @ B @ C @ F4 @ ( F2 @ I2 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F4
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X2 )
@ I6 ) ) ) ) ).
% continuous_sum
thf(fact_7522_tendsto__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ A,F2: A > B > C,A2: A > C,F4: filter @ B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ( filterlim @ B @ C @ ( F2 @ I2 ) @ ( topolo7230453075368039082e_nhds @ C @ ( A2 @ I2 ) ) @ F4 ) )
=> ( filterlim @ B @ C
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X2 )
@ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7311177749621191930dd_sum @ A @ C @ A2 @ I6 ) )
@ F4 ) ) ) ).
% tendsto_sum
thf(fact_7523_tendsto__mono,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F4: filter @ B,F11: filter @ B,F2: B > A,L: A] :
( ( ord_less_eq @ ( filter @ B ) @ F4 @ F11 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F11 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mono
thf(fact_7524_filterlim__mono,axiom,
! [B: $tType,A: $tType,F2: A > B,F24: filter @ B,F13: filter @ A,F25: filter @ B,F14: filter @ A] :
( ( filterlim @ A @ B @ F2 @ F24 @ F13 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F24 @ F25 )
=> ( ( ord_less_eq @ ( filter @ A ) @ F14 @ F13 )
=> ( filterlim @ A @ B @ F2 @ F25 @ F14 ) ) ) ) ).
% filterlim_mono
thf(fact_7525_filterlim__inf,axiom,
! [B: $tType,A: $tType,F2: A > B,F24: filter @ B,F33: filter @ B,F13: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( inf_inf @ ( filter @ B ) @ F24 @ F33 ) @ F13 )
= ( ( filterlim @ A @ B @ F2 @ F24 @ F13 )
& ( filterlim @ A @ B @ F2 @ F33 @ F13 ) ) ) ).
% filterlim_inf
thf(fact_7526_isCont__real__root,axiom,
! [X: real,N: nat] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( root @ N ) ) ).
% isCont_real_root
thf(fact_7527_continuous__at__within__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G )
=> ( ( ( G @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_7528_isCont__mult,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [A2: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( times_times @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% isCont_mult
thf(fact_7529_isCont__add,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [A2: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% isCont_add
thf(fact_7530_isCont__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% isCont_diff
thf(fact_7531_isCont__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) ) ) ) ) ).
% isCont_minus
thf(fact_7532_isCont__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [A2: A,F2: A > B,N: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N ) ) ) ) ).
% isCont_power
thf(fact_7533_continuous__at__within__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_7534_isCont__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [A3: set @ A,A2: B,F2: A > B > C] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ A2 @ ( top_top @ ( set @ B ) ) ) @ ( F2 @ X3 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ A2 @ ( top_top @ ( set @ B ) ) )
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I4: A] : ( F2 @ I4 @ X2 )
@ A3 ) ) ) ) ).
% isCont_sum
thf(fact_7535_continuous__at__within__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,S: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_sgn
thf(fact_7536_isCont__cos_H,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( cos @ B @ ( F2 @ X2 ) ) ) ) ) ).
% isCont_cos'
thf(fact_7537_isCont__sin_H,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( sin @ B @ ( F2 @ X2 ) ) ) ) ) ).
% isCont_sin'
thf(fact_7538_isCont__exp_H,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( exp @ A @ ( F2 @ X2 ) ) ) ) ) ).
% isCont_exp'
thf(fact_7539_isCont__pochhammer,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [Z3: A,N: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ Z3 @ ( top_top @ ( set @ A ) ) )
@ ^ [Z7: A] : ( comm_s3205402744901411588hammer @ A @ Z7 @ N ) ) ) ).
% isCont_pochhammer
thf(fact_7540_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X @ H ) ) @ ( F2 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_7541_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A] :
( ( has_field_derivative @ A @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X @ H ) ) @ ( F2 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_7542_lim__exp__minus__1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( filterlim @ A @ A
@ ^ [Z7: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z7 ) @ ( one_one @ A ) ) @ Z7 )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% lim_exp_minus_1
thf(fact_7543_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N3: nat,M3: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ M3 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_7544_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K: real,F2: A > B,K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ! [H4: A] :
( ( H4
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H4 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ H4 ) ) @ ( times_times @ real @ K6 @ ( real_V7770717601297561774m_norm @ A @ H4 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_7545_isCont__eq__Lb,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M8: A] :
( ! [X6: real] :
( ( ( ord_less_eq @ real @ A2 @ X6 )
& ( ord_less_eq @ real @ X6 @ B2 ) )
=> ( ord_less_eq @ A @ M8 @ ( F2 @ X6 ) ) )
& ? [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ( F2 @ X3 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_7546_isCont__eq__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M8: A] :
( ! [X6: real] :
( ( ( ord_less_eq @ real @ A2 @ X6 )
& ( ord_less_eq @ real @ X6 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X6 ) @ M8 ) )
& ? [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ( F2 @ X3 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_7547_isCont__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M8: A] :
! [X6: real] :
( ( ( ord_less_eq @ real @ A2 @ X6 )
& ( ord_less_eq @ real @ X6 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X6 ) @ M8 ) ) ) ) ) ).
% isCont_bounded
thf(fact_7548_isCont__inverse__function2,axiom,
! [A2: real,X: real,B2: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ A2 @ Z )
=> ( ( ord_less_eq @ real @ Z @ B2 )
=> ( ( G @ ( F2 @ Z ) )
= Z ) ) )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ A2 @ Z )
=> ( ( ord_less_eq @ real @ Z @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_7549_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D: A,X: A] :
( ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X @ H ) ) @ ( F2 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_7550_isCont__ln,axiom,
! [X: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( ln_ln @ real ) ) ) ).
% isCont_ln
thf(fact_7551_isCont__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ( G @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% isCont_divide
thf(fact_7552_isCont__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_sgn
thf(fact_7553_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,F4: filter @ B,A2: A] :
( ( filterlim @ A @ B @ F2 @ F4 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X2: A] : ( F2 @ ( plus_plus @ A @ X2 @ A2 ) )
@ F4
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_7554_continuous__within__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_within_tan
thf(fact_7555_continuous__within__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_within_cot
thf(fact_7556_continuous__at__within__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: C,A3: set @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A3 ) @ F2 )
=> ( ( ( cosh @ A @ ( F2 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A3 )
@ ^ [X2: C] : ( tanh @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_tanh
thf(fact_7557_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M8: A] :
( ! [X6: real] :
( ( ( ord_less_eq @ real @ A2 @ X6 )
& ( ord_less_eq @ real @ X6 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X6 ) @ M8 ) )
& ! [N5: A] :
( ( ord_less @ A @ N5 @ M8 )
=> ? [X3: real] :
( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ord_less @ A @ N5 @ ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_7558_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_7559_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_7560_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_7561_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_7562_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( F2 @ X3 ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0
thf(fact_7563_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X3: A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S )
=> ( sums @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ ( power_power @ A @ X3 @ N3 ) )
@ ( F2 @ X3 ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0_strong
thf(fact_7564_lemma__termdiff5,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_Vector_banach @ B ) )
=> ! [K: real,F2: nat > real,G: A > nat > B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ( summable @ real @ F2 )
=> ( ! [H4: A,N2: nat] :
( ( H4
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H4 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G @ H4 @ N2 ) ) @ ( times_times @ real @ ( F2 @ N2 ) @ ( real_V7770717601297561774m_norm @ A @ H4 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( suminf @ B @ ( G @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_7565_isCont__tan_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_tan'
thf(fact_7566_isCont__arcosh,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( arcosh @ real ) ) ) ).
% isCont_arcosh
thf(fact_7567_LIM__cos__div__sin,axiom,
( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( cos @ real @ X2 ) @ ( sin @ real @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( top_top @ ( set @ real ) ) ) ) ).
% LIM_cos_div_sin
thf(fact_7568_isCont__cot_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_cot'
thf(fact_7569_DERIV__inverse__function,axiom,
! [F2: real > real,D: real,G: real > real,X: real,A2: real,B2: real] :
( ( has_field_derivative @ real @ F2 @ D @ ( topolo174197925503356063within @ real @ ( G @ X ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( D
!= ( zero_zero @ real ) )
=> ( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ A2 @ Y3 )
=> ( ( ord_less @ real @ Y3 @ B2 )
=> ( ( F2 @ ( G @ Y3 ) )
= Y3 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ G )
=> ( has_field_derivative @ real @ G @ ( inverse_inverse @ real @ D ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_7570_isCont__polynom,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: A,C2: nat > A,N: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [W3: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ W3 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% isCont_polynom
thf(fact_7571_isCont__arccos,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ arccos ) ) ) ).
% isCont_arccos
thf(fact_7572_isCont__arcsin,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ arcsin ) ) ) ).
% isCont_arcsin
thf(fact_7573_isCont__powser__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [Y3: A] :
( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ Y3 @ N3 ) ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ) ).
% isCont_powser_converges_everywhere
thf(fact_7574_LIM__less__bound,axiom,
! [B2: real,X: real,F2: real > real] :
( ( ord_less @ real @ B2 @ X )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ B2 @ X ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) ) ) ) ) ).
% LIM_less_bound
thf(fact_7575_isCont__artanh,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( artanh @ real ) ) ) ) ).
% isCont_artanh
thf(fact_7576_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I4: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_7577_isCont__inverse__function,axiom,
! [D4: real,X: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z @ X ) ) @ D4 )
=> ( ( G @ ( F2 @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z @ X ) ) @ D4 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ).
% isCont_inverse_function
thf(fact_7578_GMVT_H,axiom,
! [A2: real,B2: real,F2: real > real,G: real > real,G4: real > real,F7: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ A2 @ Z )
=> ( ( ord_less_eq @ real @ Z @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ A2 @ Z )
=> ( ( ord_less_eq @ real @ Z @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) @ G ) ) )
=> ( ! [Z: real] :
( ( ord_less @ real @ A2 @ Z )
=> ( ( ord_less @ real @ Z @ B2 )
=> ( has_field_derivative @ real @ G @ ( G4 @ Z ) @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ! [Z: real] :
( ( ord_less @ real @ A2 @ Z )
=> ( ( ord_less @ real @ Z @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( F7 @ Z ) @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [C5: real] :
( ( ord_less @ real @ A2 @ C5 )
& ( ord_less @ real @ C5 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ ( G4 @ C5 ) )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A2 ) ) @ ( F7 @ C5 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_7579_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ ( suc @ I ) ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_7580_floor__has__real__derivative,axiom,
! [A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A ) )
=> ! [X: real,F2: real > A] :
( ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ F2 )
=> ( ~ ( member @ A @ ( F2 @ X ) @ ( ring_1_Ints @ A ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ A @ ( F2 @ X2 ) ) )
@ ( zero_zero @ real )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% floor_has_real_derivative
thf(fact_7581_isCont__powser_H,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ Aa )
& ( real_V3459762299906320749_field @ Aa ) )
=> ! [A2: A,F2: A > Aa,C2: nat > Aa,K6: Aa] :
( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( summable @ Aa
@ ^ [N3: nat] : ( times_times @ Aa @ ( C2 @ N3 ) @ ( power_power @ Aa @ K6 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F2 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K6 ) )
=> ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ Aa
@ ^ [N3: nat] : ( times_times @ Aa @ ( C2 @ N3 ) @ ( power_power @ Aa @ ( F2 @ X2 ) @ N3 ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_7582_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ K6 @ N3 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ A
@ ^ [N3: nat] : ( times_times @ A @ ( C2 @ N3 ) @ ( power_power @ A @ X2 @ N3 ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_7583_summable__Leibniz_I3_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( ( ord_less @ real @ ( A2 @ ( zero_zero @ nat ) ) @ ( zero_zero @ real ) )
=> ! [N9: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_7584_summable__Leibniz_I2_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( A2 @ ( zero_zero @ nat ) ) )
=> ! [N9: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_7585_tendsto__zero__mult__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ ( A2 @ N3 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_7586_tendsto__zero__mult__left__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ C2 @ ( A2 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_7587_tendsto__zero__divide__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( A2 @ N3 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_7588_isCont__Re,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,G: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( re @ ( G @ X2 ) ) ) ) ) ).
% isCont_Re
thf(fact_7589_isCont__Im,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,G: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( im @ ( G @ X2 ) ) ) ) ) ).
% isCont_Im
thf(fact_7590_approx__from__above__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [U3: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ X @ ( U3 @ N9 ) )
& ( filterlim @ nat @ A @ U3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_7591_approx__from__below__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ? [U3: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ ( U3 @ N9 ) @ X )
& ( filterlim @ nat @ A @ U3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_7592_filterlim__sequentially__Suc,axiom,
! [A: $tType,F2: nat > A,F4: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X2: nat] : ( F2 @ ( suc @ X2 ) )
@ F4
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F2 @ F4 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_7593_continuous__rabs,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( abs_abs @ real @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_rabs
thf(fact_7594_LIMSEQ__ignore__initial__segment,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,A2: A,K: nat] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_7595_LIMSEQ__offset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,K: nat,A2: A] :
( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( plus_plus @ nat @ N3 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_offset
thf(fact_7596_seq__offset__neg,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A,K: nat] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [I4: nat] : ( F2 @ ( minus_minus @ nat @ I4 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% seq_offset_neg
thf(fact_7597_continuous__cnj,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [F4: filter @ C,G: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ complex @ F4
@ ^ [X2: C] : ( cnj @ ( G @ X2 ) ) ) ) ) ).
% continuous_cnj
thf(fact_7598_continuous__complex__iff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ( ( topolo3448309680560233919inuous @ A @ complex )
= ( ^ [F9: filter @ A,F3: A > complex] :
( ( topolo3448309680560233919inuous @ A @ real @ F9
@ ^ [X2: A] : ( re @ ( F3 @ X2 ) ) )
& ( topolo3448309680560233919inuous @ A @ real @ F9
@ ^ [X2: A] : ( im @ ( F3 @ X2 ) ) ) ) ) ) ) ).
% continuous_complex_iff
thf(fact_7599_continuous__Im,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [F4: filter @ C,G: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ real @ F4
@ ^ [X2: C] : ( im @ ( G @ X2 ) ) ) ) ) ).
% continuous_Im
thf(fact_7600_continuous__arsinh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( arsinh @ real @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_arsinh
thf(fact_7601_continuous__real__root,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real,N: nat] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( root @ N @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_real_root
thf(fact_7602_continuous__real__sqrt,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( sqrt @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_real_sqrt
thf(fact_7603_LIMSEQ__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Suc
thf(fact_7604_LIMSEQ__imp__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_imp_Suc
thf(fact_7605_LIMSEQ__const__iff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [K: A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N3: nat] : K
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
= ( K = L ) ) ) ).
% LIMSEQ_const_iff
thf(fact_7606_continuous__arctan,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( arctan @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_arctan
thf(fact_7607_continuous__Re,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [F4: filter @ C,G: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ real @ F4
@ ^ [X2: C] : ( re @ ( G @ X2 ) ) ) ) ) ).
% continuous_Re
thf(fact_7608_lim__mono,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [N4: nat,X5: nat > A,Y6: nat > A,X: A,Y: A] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( ord_less_eq @ A @ ( X5 @ N2 ) @ ( Y6 @ N2 ) ) )
=> ( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y6 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% lim_mono
thf(fact_7609_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X5: nat > A,X: A,Y6: nat > A,Y: A] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y6 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ A @ ( X5 @ N2 ) @ ( Y6 @ N2 ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_7610_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L: A,M7: nat,C3: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M7 @ N2 )
=> ( ord_less_eq @ A @ ( F2 @ N2 ) @ C3 ) )
=> ( ord_less_eq @ A @ L @ C3 ) ) ) ) ).
% Lim_bounded
thf(fact_7611_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: nat > A,L: A,N4: nat,C3: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( ord_less_eq @ A @ C3 @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ A @ C3 @ L ) ) ) ) ).
% Lim_bounded2
thf(fact_7612_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X5: nat > A,X: A,A2: A] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ A @ A2 @ ( X5 @ N2 ) ) )
=> ( ord_less_eq @ A @ A2 @ X ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_7613_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X5: nat > A,X: A,A2: A] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less_eq @ A @ ( X5 @ N2 ) @ A2 ) )
=> ( ord_less_eq @ A @ X @ A2 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_7614_Sup__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: nat > A,S: set @ A,A2: A] :
( ! [N2: nat] : ( member @ A @ ( B2 @ N2 ) @ S )
=> ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ A2 @ ( complete_Sup_Sup @ A @ S ) ) ) ) ) ).
% Sup_lim
thf(fact_7615_Inf__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: nat > A,S: set @ A,A2: A] :
( ! [N2: nat] : ( member @ A @ ( B2 @ N2 ) @ S )
=> ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ S ) @ A2 ) ) ) ) ).
% Inf_lim
thf(fact_7616_summable__LIMSEQ__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ_zero
thf(fact_7617_isCont__rabs,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( abs_abs @ real @ ( F2 @ X2 ) ) ) ) ) ).
% isCont_rabs
thf(fact_7618_isCont__cnj,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,G: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ complex @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( cnj @ ( G @ X2 ) ) ) ) ) ).
% isCont_cnj
thf(fact_7619_continuous__at__within__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,S: set @ C,F2: C > real,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ S ) @ G )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ S )
@ ^ [X2: C] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_at_within_powr
thf(fact_7620_continuous__within__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,S: set @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 )
=> ( ( ( F2 @ X )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_within_ln
thf(fact_7621_mult__nat__right__at__top,axiom,
! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( filterlim @ nat @ nat
@ ^ [X2: nat] : ( times_times @ nat @ X2 @ C2 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_7622_mult__nat__left__at__top,axiom,
! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( filterlim @ nat @ nat @ ( times_times @ nat @ C2 ) @ ( at_top @ nat ) @ ( at_top @ nat ) ) ) ).
% mult_nat_left_at_top
thf(fact_7623_monoseq__convergent,axiom,
! [X5: nat > real,B3: real] :
( ( topological_monoseq @ real @ X5 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( X5 @ I2 ) ) @ B3 )
=> ~ ! [L7: real] :
~ ( filterlim @ nat @ real @ X5 @ ( topolo7230453075368039082e_nhds @ real @ L7 ) @ ( at_top @ nat ) ) ) ) ).
% monoseq_convergent
thf(fact_7624_LIMSEQ__lessThan__iff__atMost,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: ( set @ nat ) > A,X: A] :
( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( set_ord_lessThan @ nat @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( set_ord_atMost @ nat @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_lessThan_iff_atMost
thf(fact_7625_LIMSEQ__root,axiom,
( filterlim @ nat @ real
@ ^ [N3: nat] : ( root @ N3 @ ( semiring_1_of_nat @ real @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_root
thf(fact_7626_isCont__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,F2: C > real,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( ( ( F2 @ A2 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% isCont_powr
thf(fact_7627_isCont__ln_H,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ X )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_ln'
thf(fact_7628_monoseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: nat > A,X: A] :
( ( topological_monoseq @ A @ A2 )
=> ( ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ! [N9: nat] : ( ord_less_eq @ A @ ( A2 @ N9 ) @ X )
& ! [M2: nat,N9: nat] :
( ( ord_less_eq @ nat @ M2 @ N9 )
=> ( ord_less_eq @ A @ ( A2 @ M2 ) @ ( A2 @ N9 ) ) ) )
| ( ! [N9: nat] : ( ord_less_eq @ A @ X @ ( A2 @ N9 ) )
& ! [M2: nat,N9: nat] :
( ( ord_less_eq @ nat @ M2 @ N9 )
=> ( ord_less_eq @ A @ ( A2 @ N9 ) @ ( A2 @ M2 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_7629_lim__const__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A] :
( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_const_over_n
thf(fact_7630_LIMSEQ__SEQ__conv2,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,F2: A > B,L: B] :
( ! [S4: nat > A] :
( ( ! [N9: nat] :
( ( S4 @ N9 )
!= A2 )
& ( filterlim @ nat @ A @ S4 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N3: nat] : ( F2 @ ( S4 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L )
@ ( at_top @ nat ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIMSEQ_SEQ_conv2
thf(fact_7631_LIMSEQ__SEQ__conv1,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ! [S11: nat > A] :
( ( ! [N2: nat] :
( ( S11 @ N2 )
!= A2 )
& ( filterlim @ nat @ A @ S11 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N3: nat] : ( F2 @ ( S11 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_SEQ_conv1
thf(fact_7632_LIMSEQ__SEQ__conv,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,X5: A > B,L6: B] :
( ( ! [S6: nat > A] :
( ( ! [N3: nat] :
( ( S6 @ N3 )
!= A2 )
& ( filterlim @ nat @ A @ S6 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N3: nat] : ( X5 @ ( S6 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( at_top @ nat ) ) ) )
= ( filterlim @ A @ B @ X5 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIMSEQ_SEQ_conv
thf(fact_7633_lim__inverse__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_inverse_n
thf(fact_7634_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X5: nat > A,X: A,L: nat] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( X5 @ ( times_times @ nat @ N3 @ L ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_7635_telescope__summable,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) ) ) ) ) ).
% telescope_summable
thf(fact_7636_telescope__summable_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) ) ) ) ) ).
% telescope_summable'
thf(fact_7637_nested__sequence__unique,axiom,
! [F2: nat > real,G: nat > real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ ( G @ N2 ) )
=> ( ( filterlim @ nat @ real
@ ^ [N3: nat] : ( minus_minus @ real @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) )
=> ? [L3: real] :
( ! [N9: nat] : ( ord_less_eq @ real @ ( F2 @ N9 ) @ L3 )
& ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L3 ) @ ( at_top @ nat ) )
& ! [N9: nat] : ( ord_less_eq @ real @ L3 @ ( G @ N9 ) )
& ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ L3 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_7638_LIMSEQ__inverse__zero,axiom,
! [X5: nat > real] :
( ! [R: real] :
? [N5: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N5 @ N2 )
=> ( ord_less @ real @ R @ ( X5 @ N2 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( inverse_inverse @ real @ ( X5 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_7639_lim__inverse__n_H,axiom,
( filterlim @ nat @ real
@ ^ [N3: nat] : ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% lim_inverse_n'
thf(fact_7640_LIMSEQ__root__const,axiom,
! [C2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( root @ N3 @ C2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_root_const
thf(fact_7641_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim @ nat @ real
@ ^ [N3: nat] : ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_7642_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R3: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( plus_plus @ real @ R3 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R3 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_7643_sums__def,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F3: nat > A,S8: A] :
( filterlim @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S8 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def
thf(fact_7644_sums__def__le,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F3: nat > A,S8: A] :
( filterlim @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_atMost @ nat @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S8 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def_le
thf(fact_7645_continuous__at__within__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,S: set @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A2 ) )
=> ( ( ( F2 @ A2 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A2 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X2: A] : ( log @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_at_within_log
thf(fact_7646_increasing__LIMSEQ,axiom,
! [F2: nat > real,L: real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ L )
=> ( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [N9: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F2 @ N9 ) @ E ) ) )
=> ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_7647_lim__1__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_1_over_n
thf(fact_7648_LIMSEQ__Suc__n__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) @ ( semiring_1_of_nat @ A @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_7649_LIMSEQ__n__over__Suc__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_7650_telescope__sums,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
@ ( minus_minus @ A @ C2 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_7651_telescope__sums_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
@ ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ C2 ) ) ) ) ).
% telescope_sums'
thf(fact_7652_LIMSEQ__realpow__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ X ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_7653_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( divide_divide @ real @ A2 @ ( power_power @ real @ X @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_7654_LIMSEQ__abs__realpow__zero2,axiom,
! [C2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C2 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ C2 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero2
thf(fact_7655_LIMSEQ__abs__realpow__zero,axiom,
! [C2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C2 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ ( abs_abs @ real @ C2 ) ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero
thf(fact_7656_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_7657_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F3: nat > A,S8: A] :
( filterlim @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S8 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_7658_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R3: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( plus_plus @ real @ R3 @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R3 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_7659_root__test__convergence,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,X: real] :
( ( filterlim @ nat @ real
@ ^ [N3: nat] : ( root @ N3 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ nat ) )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% root_test_convergence
thf(fact_7660_summable__LIMSEQ,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( suminf @ A @ F2 ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ
thf(fact_7661_summable__LIMSEQ_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( suminf @ A @ F2 ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ'
thf(fact_7662_isCont__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A2 ) )
=> ( ( ( F2 @ A2 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A2 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( log @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ) ) ).
% isCont_log
thf(fact_7663_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A,L6: A,R3: real] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ No @ N9 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X5 @ N9 ) @ L6 ) ) @ R3 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_7664_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A,L6: A] :
( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [No2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No2 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X5 @ N2 ) @ L6 ) ) @ R ) ) )
=> ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_7665_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No3 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X5 @ N3 ) @ L6 ) ) @ R5 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_7666_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_7667_tendsto__exp__limit__sequentially,axiom,
! [X: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N3 ) ) ) @ N3 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( at_top @ nat ) ) ).
% tendsto_exp_limit_sequentially
thf(fact_7668_tendsto__at__iff__sequentially,axiom,
! [C: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > C,A2: C,X: A,S: set @ A] :
( ( filterlim @ A @ C @ F2 @ ( topolo7230453075368039082e_nhds @ C @ A2 ) @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ! [X8: nat > A] :
( ! [I4: nat] : ( member @ A @ ( X8 @ I4 ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C @ ( comp @ A @ C @ nat @ F2 @ X8 ) @ ( topolo7230453075368039082e_nhds @ C @ A2 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% tendsto_at_iff_sequentially
thf(fact_7669_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: B > nat,F4: filter @ B,X: A] :
( ( filterlim @ B @ nat @ F2 @ ( at_top @ nat ) @ F4 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y4: B] : ( power_power @ A @ X @ ( F2 @ Y4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_power_zero
thf(fact_7670_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R3: real] :
( filterlim @ nat @ real
@ ^ [N3: nat] : ( times_times @ real @ R3 @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R3 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_7671_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ! [N2: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_7672_summable__Leibniz_I1_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( summable @ real
@ ^ [N3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( A2 @ N3 ) ) ) ) ) ).
% summable_Leibniz(1)
thf(fact_7673_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Df: A,Z3: A,S: nat > A,A2: A] :
( ( has_field_derivative @ A @ F2 @ Df @ ( topolo174197925503356063within @ A @ Z3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( S @ N2 )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N3: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ Z3 @ ( S @ N3 ) ) ) @ ( F2 @ Z3 ) ) @ ( S @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( Df = A2 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_7674_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
@ ^ [N3: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_7675_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
@ ^ [N3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N3 ) @ ( power_power @ A @ X @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_7676_summable,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( summable @ real
@ ^ [N3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N3 ) @ ( A2 @ N3 ) ) ) ) ) ) ).
% summable
thf(fact_7677_cos__diff__limit__1,axiom,
! [Theta: nat > real,Theta2: real] :
( ( filterlim @ nat @ real
@ ^ [J3: nat] : ( cos @ real @ ( minus_minus @ real @ ( Theta @ J3 ) @ Theta2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ~ ! [K2: nat > int] :
~ ( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K2 @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ Theta2 )
@ ( at_top @ nat ) ) ) ).
% cos_diff_limit_1
thf(fact_7678_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim @ nat @ real
@ ^ [J3: nat] : ( cos @ real @ ( Theta @ J3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ? [K2: nat > int] :
( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K2 @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% cos_limit_1
thf(fact_7679_summable__Leibniz_I4_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(4)
thf(fact_7680_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_7681_summable__Leibniz_H_I3_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_7682_summable__Leibniz_H_I2_J,axiom,
! [A2: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_7683_sums__alternating__upper__lower,axiom,
! [A2: nat > real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ? [L3: real] :
( ! [N9: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) )
@ L3 )
& ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L3 )
@ ( at_top @ nat ) )
& ! [N9: nat] :
( ord_less_eq @ real @ L3
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L3 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_7684_summable__Leibniz_I5_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(5)
thf(fact_7685_summable__Leibniz_H_I4_J,axiom,
! [A2: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( ord_less_eq @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_7686_summable__Leibniz_H_I5_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
=> ( filterlim @ nat @ real
@ ^ [N3: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_7687_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( 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 @ ( F2 @ Y4 ) @ ( plus_plus @ B @ ( F2 @ X ) @ ( F7 @ ( minus_minus @ A @ Y4 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_7688_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,D: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ D @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ ( plus_plus @ A @ X @ H ) ) @ ( F2 @ X ) ) @ ( D @ H ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at
thf(fact_7689_bounded__linear_Ohas__derivative,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,G4: C > A,F4: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( has_derivative @ C @ A @ G @ G4 @ F4 )
=> ( has_derivative @ C @ B
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) )
@ ^ [X2: C] : ( F2 @ ( G4 @ X2 ) )
@ F4 ) ) ) ) ).
% bounded_linear.has_derivative
thf(fact_7690_bounded__linear_Ocontinuous,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,F4: filter @ C,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ A @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ B @ F4
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) ) ) ) ) ) ).
% bounded_linear.continuous
thf(fact_7691_bounded__linear_Otendsto,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,G: C > A,A2: A,F4: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) )
@ F4 ) ) ) ) ).
% bounded_linear.tendsto
thf(fact_7692_bounded__linear_Obounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ? [K9: real] :
! [X6: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X6 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X6 ) @ K9 ) ) ) ) ).
% bounded_linear.bounded
thf(fact_7693_bounded__linear__mult__left,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [Y: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ Y ) ) ) ).
% bounded_linear_mult_left
thf(fact_7694_bounded__linear__const__mult,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,X: A] :
( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X2: C] : ( times_times @ A @ X @ ( G @ X2 ) ) ) ) ) ).
% bounded_linear_const_mult
thf(fact_7695_bounded__linear__mult__const,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,Y: A] :
( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X2: C] : ( times_times @ A @ ( G @ X2 ) @ Y ) ) ) ) ).
% bounded_linear_mult_const
thf(fact_7696_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_7697_bounded__linear__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% bounded_linear_add
thf(fact_7698_bounded__linear__sub,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% bounded_linear_sub
thf(fact_7699_bounded__linear_Osuminf,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,X5: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( summable @ A @ X5 )
=> ( ( F2 @ ( suminf @ A @ X5 ) )
= ( suminf @ B
@ ^ [N3: nat] : ( F2 @ ( X5 @ N3 ) ) ) ) ) ) ) ).
% bounded_linear.suminf
thf(fact_7700_bounded__linear_Osums,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,X5: nat > A,A2: A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( sums @ A @ X5 @ A2 )
=> ( sums @ B
@ ^ [N3: nat] : ( F2 @ ( X5 @ N3 ) )
@ ( F2 @ A2 ) ) ) ) ) ).
% bounded_linear.sums
thf(fact_7701_bounded__linear_OCauchy,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,X5: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( topolo3814608138187158403Cauchy @ A @ X5 )
=> ( topolo3814608138187158403Cauchy @ B
@ ^ [N3: nat] : ( F2 @ ( X5 @ N3 ) ) ) ) ) ) ).
% bounded_linear.Cauchy
thf(fact_7702_bounded__linear__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ( real_V3181309239436604168linear @ real @ A @ ( real_Vector_of_real @ A ) ) ) ).
% bounded_linear_of_real
thf(fact_7703_bounded__linear_Osummable,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,X5: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( summable @ A @ X5 )
=> ( summable @ B
@ ^ [N3: nat] : ( F2 @ ( X5 @ N3 ) ) ) ) ) ) ).
% bounded_linear.summable
thf(fact_7704_bounded__linear__scaleR__left,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( real_V3181309239436604168linear @ real @ A
@ ^ [R5: real] : ( real_V8093663219630862766scaleR @ A @ R5 @ X ) ) ) ).
% bounded_linear_scaleR_left
thf(fact_7705_bounded__linear__const__scaleR,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > B,R3: real] :
( ( real_V3181309239436604168linear @ C @ B @ G )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ R3 @ ( G @ X2 ) ) ) ) ) ).
% bounded_linear_const_scaleR
thf(fact_7706_bounded__linear__scaleR__right,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R3: real] : ( real_V3181309239436604168linear @ A @ A @ ( real_V8093663219630862766scaleR @ A @ R3 ) ) ) ).
% bounded_linear_scaleR_right
thf(fact_7707_bounded__linear__scaleR__const,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > real,X: B] :
( ( real_V3181309239436604168linear @ C @ real @ G )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X2: C] : ( real_V8093663219630862766scaleR @ B @ ( G @ X2 ) @ X ) ) ) ) ).
% bounded_linear_scaleR_const
thf(fact_7708_bounded__linear__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) ) ) ) ) ) ).
% bounded_linear_compose
thf(fact_7709_bounded__linear__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( real_V3181309239436604168linear @ A @ A
@ ^ [X2: A] : X2 ) ) ).
% bounded_linear_ident
thf(fact_7710_bounded__linear__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [Y: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ X2 @ Y ) ) ) ).
% bounded_linear_divide
thf(fact_7711_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_7712_bounded__linear__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) ) ) ) ) ).
% bounded_linear_minus
thf(fact_7713_bounded__linear__sum,axiom,
! [I5: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [I6: set @ I5,F2: I5 > A > B] :
( ! [I2: I5] :
( ( member @ I5 @ I2 @ I6 )
=> ( real_V3181309239436604168linear @ A @ B @ ( F2 @ I2 ) ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] :
( groups7311177749621191930dd_sum @ I5 @ B
@ ^ [I4: I5] : ( F2 @ I4 @ X2 )
@ I6 ) ) ) ) ).
% bounded_linear_sum
thf(fact_7714_bounded__linear_Otendsto__zero,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,F4: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_7715_bounded__linear_OisCont,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > B,A2: C,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ B @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) ) ) ) ) ) ).
% bounded_linear.isCont
thf(fact_7716_bounded__linear_Ononneg__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ? [K9: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ K9 )
& ! [X6: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X6 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X6 ) @ K9 ) ) ) ) ) ).
% bounded_linear.nonneg_bounded
thf(fact_7717_has__derivative__within__singleton__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ G @ ( topolo174197925503356063within @ A @ X @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_V3181309239436604168linear @ A @ B @ G ) ) ) ).
% has_derivative_within_singleton_iff
thf(fact_7718_bounded__linear_Opos__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [X6: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X6 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X6 ) @ K9 ) ) ) ) ) ).
% bounded_linear.pos_bounded
thf(fact_7719_bounded__linear__intro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,K6: real] :
( ! [X3: A,Y3: A] :
( ( F2 @ ( plus_plus @ A @ X3 @ Y3 ) )
= ( plus_plus @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ! [R: real,X3: A] :
( ( F2 @ ( real_V8093663219630862766scaleR @ A @ R @ X3 ) )
= ( real_V8093663219630862766scaleR @ B @ R @ ( F2 @ X3 ) ) )
=> ( ! [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X3 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K6 ) )
=> ( real_V3181309239436604168linear @ A @ B @ F2 ) ) ) ) ) ).
% bounded_linear_intro
thf(fact_7720_has__derivative__iff__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ real
@ ^ [Y4: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y4 ) @ ( F2 @ X ) ) @ ( F7 @ ( minus_minus @ A @ Y4 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y4 @ X ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_iff_norm
thf(fact_7721_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( 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 @ ( F2 @ Y4 ) @ ( F2 @ X ) ) @ ( F7 @ ( minus_minus @ A @ Y4 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_7722_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F7: A > B,X: A,F2: A > B,S: set @ 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 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y4 ) @ ( F2 @ X ) ) @ ( F7 @ ( minus_minus @ A @ Y4 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivativeI
thf(fact_7723_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ? [E3: A > B] :
( ! [H: A] :
( ( F2 @ ( plus_plus @ A @ X @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X ) @ ( F7 @ H ) ) @ ( E3 @ H ) ) )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E3 @ H ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% has_derivative_iff_Ex
thf(fact_7724_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( 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 @ ( F2 @ Y4 ) @ ( plus_plus @ B @ ( F2 @ X ) @ ( F7 @ ( minus_minus @ A @ Y4 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_within
thf(fact_7725_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F3: A > B,F15: A > B,F9: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F15 )
& ( filterlim @ A @ B
@ ^ [Y4: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y4
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X2: A] : X2 ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F3 @ Y4 )
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X2: A] : X2 ) ) )
@ ( F15
@ ( minus_minus @ A @ Y4
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X2: A] : X2 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F9 ) ) ) ) ) ).
% has_derivative_def
thf(fact_7726_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X: A,S2: set @ A,F2: A > B,F7: A > B] :
( ( member @ A @ X @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ? [E3: A > B] :
( ! [H: A] :
( ( member @ A @ ( plus_plus @ A @ X @ H ) @ S2 )
=> ( ( F2 @ ( plus_plus @ A @ X @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X ) @ ( F7 @ H ) ) @ ( E3 @ H ) ) ) )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E3 @ H ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% has_derivative_at_within_iff_Ex
thf(fact_7727_lim__const,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A] :
( ( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat )
@ ^ [M3: nat] : A2 )
= A2 ) ) ).
% lim_const
thf(fact_7728_open__UN,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: set @ B,B3: B > ( set @ A )] :
( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( topolo1002775350975398744n_open @ A @ ( B3 @ X3 ) ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ).
% open_UN
thf(fact_7729_open__INT,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: set @ B,B3: B > ( set @ A )] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A3 )
=> ( topolo1002775350975398744n_open @ A @ ( B3 @ X3 ) ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ) ).
% open_INT
thf(fact_7730_first__countable__basis,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
? [A10: nat > ( set @ A )] :
( ! [I3: nat] :
( ( member @ A @ X @ ( A10 @ I3 ) )
& ( topolo1002775350975398744n_open @ A @ ( A10 @ I3 ) ) )
& ! [S11: set @ A] :
( ( ( topolo1002775350975398744n_open @ A @ S11 )
& ( member @ A @ X @ S11 ) )
=> ? [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( A10 @ I2 ) @ S11 ) ) ) ) ).
% first_countable_basis
thf(fact_7731_open__subopen,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S6: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S6 )
=> ? [T9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T9 )
& ( member @ A @ X2 @ T9 )
& ( ord_less_eq @ ( set @ A ) @ T9 @ S6 ) ) ) ) ) ) ).
% open_subopen
thf(fact_7732_openI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ? [T10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T10 )
& ( member @ A @ X3 @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ S2 ) ) )
=> ( topolo1002775350975398744n_open @ A @ S2 ) ) ) ).
% openI
thf(fact_7733_open__Collect__conj,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,Q: A > $o] :
( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ P ) )
=> ( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ Q ) )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ) ) ).
% open_Collect_conj
thf(fact_7734_open__Collect__disj,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,Q: A > $o] :
( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ P ) )
=> ( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ Q ) )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) ) ) ) ) ).
% open_Collect_disj
thf(fact_7735_open__Collect__const,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: $o] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X2: A] : P ) ) ) ).
% open_Collect_const
thf(fact_7736_not__open__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X: A] :
~ ( topolo1002775350975398744n_open @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% not_open_singleton
thf(fact_7737_Inf__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A3: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less @ A @ X @ X3 ) )
=> ~ ( member @ A @ ( complete_Inf_Inf @ A @ A3 ) @ A3 ) ) ) ) ).
% Inf_notin_open
thf(fact_7738_Sup__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A3: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ord_less @ A @ X3 @ X ) )
=> ~ ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ).
% Sup_notin_open
thf(fact_7739_at__within__open__subset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A,S2: set @ A,T5: set @ A] :
( ( member @ A @ A2 @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( ord_less_eq @ ( set @ A ) @ S2 @ T5 )
=> ( ( topolo174197925503356063within @ A @ A2 @ T5 )
= ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% at_within_open_subset
thf(fact_7740_open__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( ord_less @ A @ X @ Y )
=> ? [B6: A] :
( ( ord_less @ A @ X @ B6 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ B6 ) @ S2 ) ) ) ) ) ) ).
% open_right
thf(fact_7741_Lim__ident__at,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,S: set @ A] :
( ( ( topolo174197925503356063within @ A @ X @ S )
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X2: A] : X2 )
= X ) ) ) ).
% Lim_ident_at
thf(fact_7742_lim__explicit,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,F0: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ F0 ) @ ( at_top @ nat ) )
= ( ! [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
=> ( ( member @ A @ F0 @ S6 )
=> ? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( member @ A @ ( F2 @ N3 ) @ S6 ) ) ) ) ) ) ) ).
% lim_explicit
thf(fact_7743_continuous__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G )
=> ( ( ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_7744_continuous__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F4: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_inverse
thf(fact_7745_continuous__def,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ( ( topolo3448309680560233919inuous @ A @ B )
= ( ^ [F9: filter @ A,F3: A > B] :
( filterlim @ A @ B @ F3
@ ( topolo7230453075368039082e_nhds @ B
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F9
@ ^ [X2: A] : X2 ) ) )
@ F9 ) ) ) ) ).
% continuous_def
thf(fact_7746_continuous__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_sgn
thf(fact_7747_t2__space__class_OLim__def,axiom,
! [A: $tType,F: $tType] :
( ( topological_t2_space @ A )
=> ( ( topolo3827282254853284352ce_Lim @ F @ A )
= ( ^ [A4: filter @ F,F3: F > A] :
( the @ A
@ ^ [L2: A] : ( filterlim @ F @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L2 ) @ A4 ) ) ) ) ) ).
% t2_space_class.Lim_def
thf(fact_7748_at__within__nhd,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A,S2: set @ A,T5: set @ A,U4: set @ A] :
( ( member @ A @ X @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ T5 @ S2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U4 @ S2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X @ T5 )
= ( topolo174197925503356063within @ A @ X @ U4 ) ) ) ) ) ) ).
% at_within_nhd
thf(fact_7749_continuous__powr,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F4 @ G )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_powr
thf(fact_7750_continuous__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_ln
thf(fact_7751_at__eq__bot__iff,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A] :
( ( ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) )
= ( topolo1002775350975398744n_open @ A @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_eq_bot_iff
thf(fact_7752_suminf__eq__lim,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A )
= ( ^ [F3: nat > A] :
( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat )
@ ^ [N3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N3 ) ) ) ) ) ) ).
% suminf_eq_lim
thf(fact_7753_lim__def,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X5: nat > A] :
( ( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat ) @ X5 )
= ( the @ A
@ ^ [L8: A] : ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L8 ) @ ( at_top @ nat ) ) ) ) ) ).
% lim_def
thf(fact_7754_continuous__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F4 @ F2 )
=> ( ( ( cos @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F4
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_tan
thf(fact_7755_continuous__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F4 @ F2 )
=> ( ( ( sin @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F4
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_cot
thf(fact_7756_continuous__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F2 )
=> ( ( ( cosh @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ C @ C @ F4
@ ^ [X2: C] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X2: C] : ( tanh @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_tanh
thf(fact_7757_continuous__arcosh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( ord_less @ real @ ( one_one @ real )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( arcosh @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_arcosh
thf(fact_7758_continuous__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F4 @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( log @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_log
thf(fact_7759_continuous__artanh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( member @ real
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X2: A] : X2 ) )
@ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( artanh @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_artanh
thf(fact_7760_tendsto__offset__zero__iff,axiom,
! [C: $tType,D3: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D3 )
& ( zero @ C ) )
=> ! [A2: A,S2: set @ A,F2: A > D3,L6: D3] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( member @ A @ A2 @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( filterlim @ A @ D3 @ F2 @ ( topolo7230453075368039082e_nhds @ D3 @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ S2 ) )
= ( filterlim @ A @ D3
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D3 @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_7761_has__derivativeI__sandwich,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [E2: real,F7: A > B,S: set @ A,X: A,F2: A > B,H7: A > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ F7 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ S )
=> ( ( Y3 != X )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X ) @ E2 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y3 ) @ ( F2 @ X ) ) @ ( F7 @ ( minus_minus @ A @ Y3 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ X ) ) ) @ ( H7 @ Y3 ) ) ) ) )
=> ( ( filterlim @ A @ real @ H7 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivativeI_sandwich
thf(fact_7762_dist__self,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A] :
( ( real_V557655796197034286t_dist @ A @ X @ X )
= ( zero_zero @ real ) ) ) ).
% dist_self
thf(fact_7763_dist__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ( real_V557655796197034286t_dist @ A @ X @ Y )
= ( zero_zero @ real ) )
= ( X = Y ) ) ) ).
% dist_eq_0_iff
thf(fact_7764_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_7765_zero__less__dist__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) )
= ( X != Y ) ) ) ).
% zero_less_dist_iff
thf(fact_7766_dist__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) )
= ( X = Y ) ) ) ).
% dist_le_zero_iff
thf(fact_7767_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_7768_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_7769_open__ball,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,D4: real] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [Y4: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y4 ) @ D4 ) ) ) ) ).
% open_ball
thf(fact_7770_open__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S6: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S6 )
=> ? [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
& ! [Y4: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y4 @ X2 ) @ E3 )
=> ( member @ A @ Y4 @ S6 ) ) ) ) ) ) ) ).
% open_dist
thf(fact_7771_dist__triangle__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z3: A,Y: A,E2: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z3 ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z3 ) ) @ E2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E2 ) ) ) ).
% dist_triangle_lt
thf(fact_7772_dist__triangle__less__add,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,Y: A,E1: real,X22: A,E22: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ Y ) @ E1 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ Y ) @ E22 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X22 ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% dist_triangle_less_add
thf(fact_7773_dist__commute__lessI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X: A,E2: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X ) @ E2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E2 ) ) ) ).
% dist_commute_lessI
thf(fact_7774_dist__pos__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) ) ) ) ).
% dist_pos_lt
thf(fact_7775_dist__not__less__zero,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) ) ) ).
% dist_not_less_zero
thf(fact_7776_norm__conv__dist,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A )
= ( ^ [X2: A] : ( real_V557655796197034286t_dist @ A @ X2 @ ( zero_zero @ A ) ) ) ) ) ).
% norm_conv_dist
thf(fact_7777_abs__dist__diff__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [A2: A,B2: A,C2: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V557655796197034286t_dist @ A @ A2 @ B2 ) @ ( real_V557655796197034286t_dist @ A @ B2 @ C2 ) ) ) @ ( real_V557655796197034286t_dist @ A @ A2 @ C2 ) ) ) ).
% abs_dist_diff_le
thf(fact_7778_zero__le__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) ) ) ).
% zero_le_dist
thf(fact_7779_dist__triangle__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z3: A,Y: A,E2: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z3 ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z3 ) ) @ E2 )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E2 ) ) ) ).
% dist_triangle_le
thf(fact_7780_dist__triangle3,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A,A2: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ A2 @ X ) @ ( real_V557655796197034286t_dist @ A @ A2 @ Y ) ) ) ) ).
% dist_triangle3
thf(fact_7781_dist__triangle2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A,Z3: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z3 ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z3 ) ) ) ) ).
% dist_triangle2
thf(fact_7782_dist__triangle,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z3: A,Y: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z3 ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z3 ) ) ) ) ).
% dist_triangle
thf(fact_7783_continuous__dist,axiom,
! [A: $tType,D3: $tType] :
( ( ( topological_t2_space @ D3 )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F4: filter @ D3,F2: D3 > A,G: D3 > A] :
( ( topolo3448309680560233919inuous @ D3 @ A @ F4 @ F2 )
=> ( ( topolo3448309680560233919inuous @ D3 @ A @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D3 @ real @ F4
@ ^ [X2: D3] : ( real_V557655796197034286t_dist @ A @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_dist
thf(fact_7784_has__field__derivative__transform__within,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F7: A,A2: A,S2: set @ A,D4: real,G: A > A] :
( ( has_field_derivative @ A @ F2 @ F7 @ ( topolo174197925503356063within @ A @ A2 @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
=> ( ( member @ A @ A2 @ S2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ D4 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) ) )
=> ( has_field_derivative @ A @ G @ F7 @ ( topolo174197925503356063within @ A @ A2 @ S2 ) ) ) ) ) ) ) ).
% has_field_derivative_transform_within
thf(fact_7785_has__derivative__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X: A,S: set @ A,D4: real,G: A > B] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
=> ( ( member @ A @ X @ S )
=> ( ! [X16: A] :
( ( member @ A @ X16 @ S )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X ) @ D4 )
=> ( ( F2 @ X16 )
= ( G @ X16 ) ) ) )
=> ( has_derivative @ A @ B @ G @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivative_transform_within
thf(fact_7786_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M9 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_7787_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S8: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S8 @ N3 ) @ ( S8 @ N7 ) ) @ E3 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_7788_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,E2: real] :
( ( topolo3814608138187158403Cauchy @ A @ X5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M8: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M8 @ M2 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ M8 @ N9 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ M2 ) @ ( X5 @ N9 ) ) @ E2 ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_7789_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M10 @ M4 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M10 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ M4 ) @ ( X5 @ N2 ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X5 ) ) ) ).
% metric_CauchyI
thf(fact_7790_tendsto__dist,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F4: filter @ B,G: B > A,M: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ M ) @ F4 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( real_V557655796197034286t_dist @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( real_V557655796197034286t_dist @ A @ L @ M ) )
@ F4 ) ) ) ) ).
% tendsto_dist
thf(fact_7791_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,C2: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_7792_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,A2: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A2 )
=> ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_7793_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,A2: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A2 )
=> ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_7794_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,C2: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_7795_dist__triangle__half__r,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X15: A,E2: real,X22: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X15 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X22 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X22 ) @ E2 ) ) ) ) ).
% dist_triangle_half_r
thf(fact_7796_dist__triangle__half__l,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,Y: A,E2: real,X22: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ Y ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ Y ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X22 ) @ E2 ) ) ) ) ).
% dist_triangle_half_l
thf(fact_7797_metric__LIM__imp__LIM,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F2: C > A,L: A,A2: C,G: C > B,M: B] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) )
=> ( ! [X3: C] :
( ( X3 != A2 )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G @ X3 ) @ M ) @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X3 ) @ L ) ) )
=> ( filterlim @ C @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ M ) @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) ) ) ) ) ).
% metric_LIM_imp_LIM
thf(fact_7798_Lim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L: B,X: A,S2: set @ A,D4: real,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
=> ( ! [X16: A] :
( ( member @ A @ X16 @ S2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X16 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X ) @ D4 )
=> ( ( F2 @ X16 )
= ( G @ X16 ) ) ) ) )
=> ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% Lim_transform_within
thf(fact_7799_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,X22: A,E2: real,X32: A,X42: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X22 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ X32 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X32 @ X42 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X42 ) @ E2 ) ) ) ) ) ).
% dist_triangle_third
thf(fact_7800_filterlim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [G: A > B,G6: filter @ B,X: A,S2: set @ A,F4: filter @ B,D4: real,F2: A > B] :
( ( filterlim @ A @ B @ G @ G6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ G6 @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
=> ( ! [X16: A] :
( ( member @ A @ X16 @ S2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X16 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X ) @ D4 )
=> ( ( F2 @ X16 )
= ( G @ X16 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ F4 @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ) ).
% filterlim_transform_within
thf(fact_7801_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F3: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ M3 ) @ ( F3 @ N3 ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_7802_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M10: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M10 @ M4 )
=> ! [N2: nat] :
( ( ord_less @ nat @ M4 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ M4 ) @ ( X5 @ N2 ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X5 ) ) ) ).
% CauchyI'
thf(fact_7803_dist__of__nat,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [M: nat,N: nat] :
( ( real_V557655796197034286t_dist @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ring_1_of_int @ real @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) ) ) ) ) ).
% dist_of_nat
thf(fact_7804_tendsto__dist__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V557655796197034286t_dist @ B @ ( F2 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_dist_iff
thf(fact_7805_LIM__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F2: A > B,L6: B,A2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S8 )
& ! [X2: A] :
( ( ( X2 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A2 ) @ S8 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X2 ) @ L6 ) @ R5 ) ) ) ) ) ) ) ).
% LIM_def
thf(fact_7806_metric__LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F2: A > B,L6: B,A2: A,R3: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [S3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S3 )
& ! [X6: A] :
( ( ( X6 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X6 @ A2 ) @ S3 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X6 ) @ L6 ) @ R3 ) ) ) ) ) ) ).
% metric_LIM_D
thf(fact_7807_metric__LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [A2: A,F2: A > B,L6: B] :
( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [S10: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S10 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ S10 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X3 ) @ L6 ) @ R ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% metric_LIM_I
thf(fact_7808_metric__LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G: A > B,L: B,A2: A,R2: real,F2: A > B] :
( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ! [X3: A] :
( ( X3 != A2 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ R2 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_equal2
thf(fact_7809_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No3 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ N3 ) @ L6 ) @ R5 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_7810_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,L6: A] :
( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [No2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No2 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ N2 ) @ L6 ) @ R ) ) )
=> ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_7811_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,L6: A,R3: real] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ No @ N9 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ N9 ) @ L6 ) @ R3 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_7812_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_7813_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X8: nat > A] :
! [J3: nat] :
? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M9 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_7814_metric__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,B2: B,A2: A,G: B > C,C2: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ D2 ) )
=> ( ( F2 @ X3 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_compose2
thf(fact_7815_metric__isCont__LIM__compose2,axiom,
! [D3: $tType,C: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ D3 ) )
=> ! [A2: A,F2: A > C,G: C > D3,L: D3] :
( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( filterlim @ C @ D3 @ G @ ( topolo7230453075368039082e_nhds @ D3 @ L ) @ ( topolo174197925503356063within @ C @ ( F2 @ A2 ) @ ( top_top @ ( set @ C ) ) ) )
=> ( ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ D2 ) )
=> ( ( F2 @ X3 )
!= ( F2 @ A2 ) ) ) )
=> ( filterlim @ A @ D3
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ D3 @ L )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_isCont_LIM_compose2
thf(fact_7816_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ No3 )
& ! [N3: nat] :
( ( ord_less_eq @ nat @ No3 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X5 @ N3 ) @ L6 ) @ R5 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_7817_LIM__offset__zero__iff,axiom,
! [C: $tType,D3: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D3 )
& ( zero @ C ) )
=> ! [A2: A,F2: A > D3,L6: D3] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( filterlim @ A @ D3 @ F2 @ ( topolo7230453075368039082e_nhds @ D3 @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D3
@ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D3 @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_7818_totally__bounded__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S6: set @ A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [K3: set @ A] :
( ( finite_finite2 @ A @ K3 )
& ( ord_less_eq @ ( set @ A ) @ S6
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image @ A @ ( set @ A )
@ ^ [X2: A] :
( collect @ A
@ ^ [Y4: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y4 ) @ E3 ) )
@ K3 ) ) ) ) ) ) ) ) ).
% totally_bounded_metric
thf(fact_7819_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_7820_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
= ( ord_less @ A @ K @ I ) ) ) ).
% greaterThan_iff
thf(fact_7821_cInf__greaterThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A )
& ( no_top @ A ) )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_greaterThan @ A @ X ) )
= X ) ) ).
% cInf_greaterThan
thf(fact_7822_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_7823_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_7824_at__within__Icc__at__right,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( topolo174197925503356063within @ A @ A2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ).
% at_within_Icc_at_right
thf(fact_7825_totally__bounded__subset,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [S2: set @ A,T5: set @ A] :
( ( topolo6688025880775521714ounded @ A @ S2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T5 @ S2 )
=> ( topolo6688025880775521714ounded @ A @ T5 ) ) ) ) ).
% totally_bounded_subset
thf(fact_7826_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ).
% greaterThan_def
thf(fact_7827_filterlim__at__left__to__right,axiom,
! [A: $tType,F2: real > A,F4: filter @ A,A2: real] :
( ( filterlim @ real @ A @ F2 @ F4 @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F2 @ ( uminus_uminus @ real @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A2 ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ A2 ) ) ) ) ) ).
% filterlim_at_left_to_right
thf(fact_7828_less__separate,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [A6: A,B6: A] :
( ( member @ A @ X @ ( set_ord_lessThan @ A @ A6 ) )
& ( member @ A @ Y @ ( set_ord_greaterThan @ A @ B6 ) )
& ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A6 ) @ ( set_ord_greaterThan @ A @ B6 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% less_separate
thf(fact_7829_filterlim__at__right__to__0,axiom,
! [A: $tType,F2: real > A,F4: filter @ A,A2: real] :
( ( filterlim @ real @ A @ F2 @ F4 @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F2 @ ( plus_plus @ real @ X2 @ A2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_right_to_0
thf(fact_7830_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: B > A,P5: A,F13: filter @ B,C2: A,L: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ P5 @ ( set_ord_greaterThan @ A @ P5 ) ) @ F13 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( L
= ( times_times @ A @ C2 @ P5 ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo174197925503356063within @ A @ L @ ( set_ord_greaterThan @ A @ L ) )
@ F13 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_7831_tendsto__arcosh__at__left__1,axiom,
filterlim @ real @ real @ ( arcosh @ real ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( one_one @ real ) @ ( set_ord_greaterThan @ real @ ( one_one @ real ) ) ) ).
% tendsto_arcosh_at_left_1
thf(fact_7832_isCont__If__ge,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,G: A > B,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) @ G )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( G @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( if @ B @ ( ord_less_eq @ A @ X2 @ A2 ) @ ( G @ X2 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_If_ge
thf(fact_7833_at__within__order,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,S: set @ A] :
( ( ( top_top @ ( set @ A ) )
!= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X @ S )
= ( inf_inf @ ( filter @ A )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A5 ) @ S ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_greaterThan @ A @ X ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A5 ) @ S ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_lessThan @ A @ X ) ) ) ) ) ) ) ).
% at_within_order
thf(fact_7834_filterlim__tan__at__right,axiom,
filterlim @ real @ real @ ( tan @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% filterlim_tan_at_right
thf(fact_7835_principal__le__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( principal @ A @ A3 ) @ ( principal @ A @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% principal_le_iff
thf(fact_7836_SUP__principal,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),I6: set @ B] :
( ( complete_Sup_Sup @ ( filter @ A )
@ ( image @ B @ ( filter @ A )
@ ^ [I4: B] : ( principal @ A @ ( A3 @ I4 ) )
@ I6 ) )
= ( principal @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) ) ) ).
% SUP_principal
thf(fact_7837_nhds__discrete,axiom,
! [A: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X2: A] : ( principal @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% nhds_discrete
thf(fact_7838_at__bot__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atMost @ A @ K3 ) )
@ ( set_ord_atMost @ A @ C2 ) ) ) ) ) ).
% at_bot_sub
thf(fact_7839_at__bot__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atMost @ A @ K3 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_bot_def
thf(fact_7840_filterlim__If,axiom,
! [B: $tType,A: $tType,F2: A > B,G6: filter @ B,F4: filter @ A,P: A > $o,G: A > B] :
( ( filterlim @ A @ B @ F2 @ G6 @ ( inf_inf @ ( filter @ A ) @ F4 @ ( principal @ A @ ( collect @ A @ P ) ) ) )
=> ( ( filterlim @ A @ B @ G @ G6
@ ( inf_inf @ ( filter @ A ) @ F4
@ ( principal @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( if @ B @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ G6
@ F4 ) ) ) ).
% filterlim_If
thf(fact_7841_nhds__def,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [A5: A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image @ ( set @ A ) @ ( filter @ A ) @ ( principal @ A )
@ ( collect @ ( set @ A )
@ ^ [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
& ( member @ A @ A5 @ S6 ) ) ) ) ) ) ) ) ).
% nhds_def
thf(fact_7842_filterlim__base,axiom,
! [B: $tType,A: $tType,E4: $tType,D3: $tType,C: $tType,J5: set @ A,I: A > C,I6: set @ C,F4: C > ( set @ D3 ),F2: D3 > E4,G6: A > ( set @ E4 )] :
( ! [M4: A,X3: B] :
( ( member @ A @ M4 @ J5 )
=> ( member @ C @ ( I @ M4 ) @ I6 ) )
=> ( ! [M4: A,X3: D3] :
( ( member @ A @ M4 @ J5 )
=> ( ( member @ D3 @ X3 @ ( F4 @ ( I @ M4 ) ) )
=> ( member @ E4 @ ( F2 @ X3 ) @ ( G6 @ M4 ) ) ) )
=> ( filterlim @ D3 @ E4 @ F2
@ ( complete_Inf_Inf @ ( filter @ E4 )
@ ( image @ A @ ( filter @ E4 )
@ ^ [J3: A] : ( principal @ E4 @ ( G6 @ J3 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ D3 )
@ ( image @ C @ ( filter @ D3 )
@ ^ [I4: C] : ( principal @ D3 @ ( F4 @ I4 ) )
@ I6 ) ) ) ) ) ).
% filterlim_base
thf(fact_7843_tendsto__principal__singleton,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,X: B] : ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( F2 @ X ) ) @ ( principal @ B @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% tendsto_principal_singleton
thf(fact_7844_nhds__discrete__open,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A] :
( ( topolo1002775350975398744n_open @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( topolo7230453075368039082e_nhds @ A @ X )
= ( principal @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% nhds_discrete_open
thf(fact_7845_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_7846_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_7847_exp__at__bot,axiom,
filterlim @ real @ real @ ( exp @ real ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_bot @ real ) ).
% exp_at_bot
thf(fact_7848_filterlim__inverse__at__bot__neg,axiom,
filterlim @ real @ real @ ( inverse_inverse @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_lessThan @ real @ ( zero_zero @ real ) ) ) ).
% filterlim_inverse_at_bot_neg
thf(fact_7849_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D3: $tType,I6: set @ A,F4: A > ( set @ B ),F2: B > C,G6: D3 > ( set @ C ),J5: set @ D3] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I6 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F4 @ I2 ) @ ( F4 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F4 @ J2 ) @ ( F4 @ I2 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F2
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image @ D3 @ ( filter @ C )
@ ^ [J3: D3] : ( principal @ C @ ( G6 @ J3 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image @ A @ ( filter @ B )
@ ^ [I4: A] : ( principal @ B @ ( F4 @ I4 ) )
@ I6 ) ) )
= ( ! [X2: D3] :
( ( member @ D3 @ X2 @ J5 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ I6 )
& ! [Z7: B] :
( ( member @ B @ Z7 @ ( F4 @ Y4 ) )
=> ( member @ C @ ( F2 @ Z7 ) @ ( G6 @ X2 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_7850_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X5: set @ A,F2: A > ( set @ B )] :
( ( finite_finite2 @ A @ X5 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image @ A @ ( filter @ B )
@ ^ [X2: A] : ( principal @ B @ ( F2 @ X2 ) )
@ X5 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ X5 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_7851_filterlim__tendsto__pos__mult__at__bot,axiom,
! [A: $tType,F2: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_bot @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_bot
thf(fact_7852_INT__greaterThan__UNIV,axiom,
( ( complete_Inf_Inf @ ( set @ nat ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_greaterThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% INT_greaterThan_UNIV
thf(fact_7853_ln__at__0,axiom,
filterlim @ real @ real @ ( ln_ln @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ).
% ln_at_0
thf(fact_7854_tendsto__at__botI__sequentially,axiom,
! [B: $tType] :
( ( topolo3112930676232923870pology @ B )
=> ! [F2: real > B,Y: B] :
( ! [X10: nat > real] :
( ( filterlim @ nat @ real @ X10 @ ( at_bot @ real ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ B
@ ^ [N3: nat] : ( F2 @ ( X10 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y )
@ ( at_top @ nat ) ) )
=> ( filterlim @ real @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ Y ) @ ( at_bot @ real ) ) ) ) ).
% tendsto_at_botI_sequentially
thf(fact_7855_at__within__def,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [A5: A,S8: set @ A] : ( inf_inf @ ( filter @ A ) @ ( topolo7230453075368039082e_nhds @ A @ A5 ) @ ( principal @ A @ ( minus_minus @ ( set @ A ) @ S8 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% at_within_def
thf(fact_7856_nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X2: A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image @ real @ ( filter @ A )
@ ^ [E3: real] :
( principal @ A
@ ( collect @ A
@ ^ [Y4: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y4 @ X2 ) @ E3 ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% nhds_metric
thf(fact_7857_at__left__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ A5 @ X ) )
@ ( set_ord_lessThan @ A @ X ) ) ) ) ) ) ).
% at_left_eq
thf(fact_7858_at__right__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ X @ A5 ) )
@ ( set_ord_greaterThan @ A @ X ) ) ) ) ) ) ).
% at_right_eq
thf(fact_7859_nhds__order,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X2: A] :
( inf_inf @ ( filter @ A )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_ord_lessThan @ A @ A5 ) )
@ ( set_ord_greaterThan @ A @ X2 ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_ord_greaterThan @ A @ A5 ) )
@ ( set_ord_lessThan @ A @ X2 ) ) ) ) ) ) ) ).
% nhds_order
thf(fact_7860_at__within__eq,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [X2: A,S8: set @ A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image @ ( set @ A ) @ ( filter @ A )
@ ^ [S6: set @ A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S6 @ S8 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
& ( member @ A @ X2 @ S6 ) ) ) ) ) ) ) ) ).
% at_within_eq
thf(fact_7861_DERIV__pos__imp__increasing__at__bot,axiom,
! [B2: real,F2: real > real,Flim: real] :
( ! [X3: real] :
( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [Y2: real] :
( ( has_field_derivative @ real @ F2 @ Y2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y2 ) ) )
=> ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_bot @ real ) )
=> ( ord_less @ real @ Flim @ ( F2 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_7862_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F2: real > real,F4: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F4 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( power_power @ real @ ( F2 @ X2 ) @ N )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_7863_tendsto__arctan__at__bot,axiom,
filterlim @ real @ real @ arctan @ ( topolo7230453075368039082e_nhds @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) @ ( at_bot @ real ) ).
% tendsto_arctan_at_bot
thf(fact_7864_filterlim__pow__at__bot__even,axiom,
! [N: nat,F2: real > real,F4: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F4 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( power_power @ real @ ( F2 @ X2 ) @ N )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_7865_filterlim__tan__at__left,axiom,
filterlim @ real @ real @ ( tan @ real ) @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( set_ord_lessThan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% filterlim_tan_at_left
thf(fact_7866_exp__at__top,axiom,
filterlim @ real @ real @ ( exp @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% exp_at_top
thf(fact_7867_sqrt__at__top,axiom,
filterlim @ real @ real @ sqrt @ ( at_top @ real ) @ ( at_top @ real ) ).
% sqrt_at_top
thf(fact_7868_ln__at__top,axiom,
filterlim @ real @ real @ ( ln_ln @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% ln_at_top
thf(fact_7869_filterlim__at__top__add__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( plus_plus @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_top_add_at_top
thf(fact_7870_filterlim__at__top__mult__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_top_mult_at_top
thf(fact_7871_filterlim__tendsto__add__at__top,axiom,
! [A: $tType,F2: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( plus_plus @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_tendsto_add_at_top
thf(fact_7872_filterlim__real__sequentially,axiom,
filterlim @ nat @ real @ ( semiring_1_of_nat @ real ) @ ( at_top @ real ) @ ( at_top @ nat ) ).
% filterlim_real_sequentially
thf(fact_7873_filterlim__uminus__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X2: A] : ( uminus_uminus @ real @ ( F2 @ X2 ) )
@ ( at_bot @ real )
@ F4 ) ) ).
% filterlim_uminus_at_top
thf(fact_7874_filterlim__uminus__at__bot,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X2: A] : ( uminus_uminus @ real @ ( F2 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ).
% filterlim_uminus_at_bot
thf(fact_7875_filterlim__at__top__mirror,axiom,
! [A: $tType,F2: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F4 @ ( at_top @ real ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F2 @ ( uminus_uminus @ real @ X2 ) )
@ F4
@ ( at_bot @ real ) ) ) ).
% filterlim_at_top_mirror
thf(fact_7876_filterlim__at__bot__mirror,axiom,
! [A: $tType,F2: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F4 @ ( at_bot @ real ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F2 @ ( uminus_uminus @ real @ X2 ) )
@ F4
@ ( at_top @ real ) ) ) ).
% filterlim_at_bot_mirror
thf(fact_7877_filterlim__uminus__at__top__at__bot,axiom,
filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_top @ real ) @ ( at_bot @ real ) ).
% filterlim_uminus_at_top_at_bot
thf(fact_7878_filterlim__uminus__at__bot__at__top,axiom,
filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_bot @ real ) @ ( at_top @ real ) ).
% filterlim_uminus_at_bot_at_top
thf(fact_7879_filterlim__pow__at__top,axiom,
! [A: $tType,N: nat,F2: A > real,F4: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( power_power @ real @ ( F2 @ X2 ) @ N )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_pow_at_top
thf(fact_7880_real__tendsto__divide__at__top,axiom,
! [A: $tType,F2: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% real_tendsto_divide_at_top
thf(fact_7881_tendsto__inverse__0__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ).
% tendsto_inverse_0_at_top
thf(fact_7882_filterlim__sequentially__iff__filterlim__real,axiom,
! [A: $tType,F2: A > nat,F4: filter @ A] :
( ( filterlim @ A @ nat @ F2 @ ( at_top @ nat ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X2: A] : ( semiring_1_of_nat @ real @ ( F2 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ).
% filterlim_sequentially_iff_filterlim_real
thf(fact_7883_filterlim__tendsto__pos__mult__at__top,axiom,
! [A: $tType,F2: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_top
thf(fact_7884_filterlim__at__top__mult__tendsto__pos,axiom,
! [A: $tType,F2: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( G @ X2 ) @ ( F2 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_at_top_mult_tendsto_pos
thf(fact_7885_tendsto__neg__powr,axiom,
! [A: $tType,S: real,F2: A > real,F4: filter @ A] :
( ( ord_less @ real @ S @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ S )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_neg_powr
thf(fact_7886_ln__x__over__x__tendsto__0,axiom,
( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( ln_ln @ real @ X2 ) @ X2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% ln_x_over_x_tendsto_0
thf(fact_7887_filterlim__at__top__to__right,axiom,
! [A: $tType,F2: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F4 @ ( at_top @ real ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F2 @ ( inverse_inverse @ real @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_top_to_right
thf(fact_7888_filterlim__at__right__to__top,axiom,
! [A: $tType,F2: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F4 @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F2 @ ( inverse_inverse @ real @ X2 ) )
@ F4
@ ( at_top @ real ) ) ) ).
% filterlim_at_right_to_top
thf(fact_7889_filterlim__inverse__at__top__right,axiom,
filterlim @ real @ real @ ( inverse_inverse @ real ) @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ).
% filterlim_inverse_at_top_right
thf(fact_7890_filterlim__inverse__at__right__top,axiom,
filterlim @ real @ real @ ( inverse_inverse @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) @ ( at_top @ real ) ).
% filterlim_inverse_at_right_top
thf(fact_7891_tendsto__at__topI__sequentially,axiom,
! [B: $tType] :
( ( topolo3112930676232923870pology @ B )
=> ! [F2: real > B,Y: B] :
( ! [X10: nat > real] :
( ( filterlim @ nat @ real @ X10 @ ( at_top @ real ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ B
@ ^ [N3: nat] : ( F2 @ ( X10 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y )
@ ( at_top @ nat ) ) )
=> ( filterlim @ real @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ Y ) @ ( at_top @ real ) ) ) ) ).
% tendsto_at_topI_sequentially
thf(fact_7892_filterlim__tendsto__neg__mult__at__bot,axiom,
! [A: $tType,F2: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( times_times @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_neg_mult_at_bot
thf(fact_7893_tendsto__power__div__exp__0,axiom,
! [K: nat] :
( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( power_power @ real @ X2 @ K ) @ ( exp @ real @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% tendsto_power_div_exp_0
thf(fact_7894_tendsto__exp__limit__at__top,axiom,
! [X: real] :
( filterlim @ real @ real
@ ^ [Y4: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ Y4 ) ) @ Y4 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( at_top @ real ) ) ).
% tendsto_exp_limit_at_top
thf(fact_7895_tendsto__arctan__at__top,axiom,
filterlim @ real @ real @ arctan @ ( topolo7230453075368039082e_nhds @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( at_top @ real ) ).
% tendsto_arctan_at_top
thf(fact_7896_DERIV__neg__imp__decreasing__at__top,axiom,
! [B2: real,F2: real > real,Flim: real] :
( ! [X3: real] :
( ( ord_less_eq @ real @ B2 @ X3 )
=> ? [Y2: real] :
( ( has_field_derivative @ real @ F2 @ Y2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y2 @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_top @ real ) )
=> ( ord_less @ real @ Flim @ ( F2 @ B2 ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_7897_at__infinity__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( at_infinity @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ real @ ( filter @ A )
@ ^ [R5: real] :
( principal @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ X2 ) ) ) )
@ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% at_infinity_def
thf(fact_7898_lhopital__left__at__top,axiom,
! [G: real > real,X: real,G4: real > real,F2: real > real,F7: real > real,Y: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top
thf(fact_7899_eventually__const,axiom,
! [A: $tType,F4: filter @ A,P: $o] :
( ( F4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X2: A] : P
@ F4 )
= P ) ) ).
% eventually_const
thf(fact_7900_eventually__at__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,Y: A,P: A > $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ X @ B5 )
& ! [Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ( ord_less @ A @ Y4 @ B5 )
=> ( P @ Y4 ) ) ) ) ) ) ) ) ).
% eventually_at_right
thf(fact_7901_eventually__at__right__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ X @ B5 )
& ! [Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ( ord_less @ A @ Y4 @ B5 )
=> ( P @ Y4 ) ) ) ) ) ) ) ).
% eventually_at_right_field
thf(fact_7902_eventually__at__right__less,axiom,
! [A: $tType] :
( ( ( no_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A] : ( eventually @ A @ ( ord_less @ A @ X ) @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) ) ) ).
% eventually_at_right_less
thf(fact_7903_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_7904_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N7: A] :
! [N3: A] :
( ( ord_less @ A @ N3 @ N7 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_7905_eventually__at__bot__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : X2 != C2
@ ( at_bot @ A ) ) ) ).
% eventually_at_bot_not_equal
thf(fact_7906_at__bot__le__at__infinity,axiom,
ord_less_eq @ ( filter @ real ) @ ( at_bot @ real ) @ ( at_infinity @ real ) ).
% at_bot_le_at_infinity
thf(fact_7907_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N7: A] :
! [N3: A] :
( ( ord_less_eq @ A @ N3 @ N7 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_7908_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_7909_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F4 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ Z9 )
@ F4 ) ) ) ) ).
% filterlim_at_bot
thf(fact_7910_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F4 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X2 ) )
@ F4 ) ) ) ) ).
% filterlim_at_top
thf(fact_7911_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F4 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ Z9 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_7912_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F4 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X2 ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_7913_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,F4: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( at_top @ A ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F4 )
=> ( filterlim @ B @ A @ G @ ( at_top @ A ) @ F4 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_7914_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A] :
( ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F2 @ ( G @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( at_top @ A ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_7915_eventually__nhds__within__iff__sequentially,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [P: A > $o,A2: A,S: set @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( principal @ A @ S ) ) )
= ( ! [F3: nat > A] :
( ( ! [N3: nat] : ( member @ A @ ( F3 @ N3 ) @ S )
& ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( eventually @ nat
@ ^ [N3: nat] : ( P @ ( F3 @ N3 ) )
@ ( at_top @ nat ) ) ) ) ) ) ).
% eventually_nhds_within_iff_sequentially
thf(fact_7916_sequentially__imp__eventually__nhds__within,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [S: set @ A,A2: A,P: A > $o] :
( ! [F5: nat > A] :
( ( ! [N9: nat] : ( member @ A @ ( F5 @ N9 ) @ S )
& ( filterlim @ nat @ A @ F5 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( eventually @ nat
@ ^ [N3: nat] : ( P @ ( F5 @ N3 ) )
@ ( at_top @ nat ) ) )
=> ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( principal @ A @ S ) ) ) ) ) ).
% sequentially_imp_eventually_nhds_within
thf(fact_7917_eventually__nhds__iff__sequentially,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [P: A > $o,A2: A] :
( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ A2 ) )
= ( ! [F3: nat > A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N3: nat] : ( P @ ( F3 @ N3 ) )
@ ( at_top @ nat ) ) ) ) ) ) ).
% eventually_nhds_iff_sequentially
thf(fact_7918_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F4 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ Z9 )
@ F4 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_7919_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F4 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ Z9 @ ( F2 @ X2 ) )
@ F4 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_7920_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_infinity @ real ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( F2 @ X2 ) @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F4 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_bot
thf(fact_7921_filterlim__at__infinity__imp__filterlim__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_infinity @ real ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F4 )
=> ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_top
thf(fact_7922_sequentially__imp__eventually__at,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [A2: A,P: A > $o] :
( ! [F5: nat > A] :
( ( ! [N9: nat] :
( ( F5 @ N9 )
!= A2 )
& ( filterlim @ nat @ A @ F5 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( eventually @ nat
@ ^ [N3: nat] : ( P @ ( F5 @ N3 ) )
@ ( at_top @ nat ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% sequentially_imp_eventually_at
thf(fact_7923_sequentially__imp__eventually__within,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [S: set @ A,A2: A,P: A > $o] :
( ! [F5: nat > A] :
( ( ! [N9: nat] :
( ( member @ A @ ( F5 @ N9 ) @ S )
& ( ( F5 @ N9 )
!= A2 ) )
& ( filterlim @ nat @ A @ F5 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( eventually @ nat
@ ^ [N3: nat] : ( P @ ( F5 @ N3 ) )
@ ( at_top @ nat ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S ) ) ) ) ).
% sequentially_imp_eventually_within
thf(fact_7924_eventually__INF1,axiom,
! [B: $tType,A: $tType,I: A,I6: set @ A,P: B > $o,F4: A > ( filter @ B )] :
( ( member @ A @ I @ I6 )
=> ( ( eventually @ B @ P @ ( F4 @ I ) )
=> ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F4 @ I6 ) ) ) ) ) ).
% eventually_INF1
thf(fact_7925_eventually__gt__at__top,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] : ( eventually @ A @ ( ord_less @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_gt_at_top
thf(fact_7926_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N7: A] :
! [N3: A] :
( ( ord_less @ A @ N7 @ N3 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_7927_eventually__at__top__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : X2 != C2
@ ( at_top @ A ) ) ) ).
% eventually_at_top_not_equal
thf(fact_7928_at__top__le__at__infinity,axiom,
ord_less_eq @ ( filter @ real ) @ ( at_top @ real ) @ ( at_infinity @ real ) ).
% at_top_le_at_infinity
thf(fact_7929_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N7: A] :
! [N3: A] :
( ( ord_less_eq @ A @ N7 @ N3 )
=> ( P @ N3 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_7930_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,P: A > $o] :
( ! [X3: A] :
( ( ord_less_eq @ A @ C2 @ X3 )
=> ( P @ X3 ) )
=> ( eventually @ A @ P @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_7931_eventually__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] : ( eventually @ A @ ( ord_less_eq @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_ge_at_top
thf(fact_7932_eventually__Inf__base,axiom,
! [A: $tType,B3: set @ ( filter @ A ),P: A > $o] :
( ( B3
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F6: filter @ A] :
( ( member @ ( filter @ A ) @ F6 @ B3 )
=> ! [G3: filter @ A] :
( ( member @ ( filter @ A ) @ G3 @ B3 )
=> ? [X6: filter @ A] :
( ( member @ ( filter @ A ) @ X6 @ B3 )
& ( ord_less_eq @ ( filter @ A ) @ X6 @ ( inf_inf @ ( filter @ A ) @ F6 @ G3 ) ) ) ) )
=> ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B3 ) )
= ( ? [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ B3 )
& ( eventually @ A @ P @ X2 ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_7933_eventually__nhds__in__open,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ X @ S )
=> ( eventually @ A
@ ^ [Y4: A] : ( member @ A @ Y4 @ S )
@ ( topolo7230453075368039082e_nhds @ A @ X ) ) ) ) ) ).
% eventually_nhds_in_open
thf(fact_7934_eventually__at__filter,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,A2: A,S: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S ) )
= ( eventually @ A
@ ^ [X2: A] :
( ( X2 != A2 )
=> ( ( member @ A @ X2 @ S )
=> ( P @ X2 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 ) ) ) ) ).
% eventually_at_filter
thf(fact_7935_has__derivative__transform__eventually,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F7: A > B,X: A,S: set @ A,G: A > B] :
( ( has_derivative @ A @ B @ F2 @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( eventually @ A
@ ^ [X9: A] :
( ( F2 @ X9 )
= ( G @ X9 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
= ( G @ X ) )
=> ( ( member @ A @ X @ S )
=> ( has_derivative @ A @ B @ G @ F7 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivative_transform_eventually
thf(fact_7936_eventually__nhds__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,P: A > $o] :
( ( ord_less @ A @ B2 @ ( top_top @ A ) )
=> ( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ ( top_top @ A ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ ( top_top @ A ) )
& ! [Z7: A] :
( ( ord_less @ A @ B5 @ Z7 )
=> ( P @ Z7 ) ) ) ) ) ) ) ).
% eventually_nhds_top
thf(fact_7937_eventually__eventually,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P: A > $o,X: A] :
( ( eventually @ A
@ ^ [Y4: A] : ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ Y4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
= ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ X ) ) ) ) ).
% eventually_eventually
thf(fact_7938_t1__space__nhds,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( eventually @ A
@ ^ [X2: A] : X2 != Y
@ ( topolo7230453075368039082e_nhds @ A @ X ) ) ) ) ).
% t1_space_nhds
thf(fact_7939_has__field__derivative__cong__eventually,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,G: A > A,X: A,S2: set @ A,U2: A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ( F2 @ X )
= ( G @ X ) )
=> ( ( has_field_derivative @ A @ F2 @ U2 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
= ( has_field_derivative @ A @ G @ U2 @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% has_field_derivative_cong_eventually
thf(fact_7940_eventually__at__infinity__pos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P5: A > $o] :
( ( eventually @ A @ P5 @ ( at_infinity @ A ) )
= ( ? [B5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B5 )
& ! [X2: A] :
( ( ord_less_eq @ real @ B5 @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( P5 @ X2 ) ) ) ) ) ) ).
% eventually_at_infinity_pos
thf(fact_7941_not__eventually__impI,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ~ ( eventually @ A @ Q @ F4 )
=> ~ ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F4 ) ) ) ).
% not_eventually_impI
thf(fact_7942_eventually__conj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ F4 )
= ( ( eventually @ A @ P @ F4 )
& ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_conj_iff
thf(fact_7943_eventually__rev__mp,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F4 )
=> ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_rev_mp
thf(fact_7944_eventually__subst,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [N3: A] :
( ( P @ N3 )
= ( Q @ N3 ) )
@ F4 )
=> ( ( eventually @ A @ P @ F4 )
= ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_subst
thf(fact_7945_eventually__elim2,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o,R2: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A @ Q @ F4 )
=> ( ! [I2: A] :
( ( P @ I2 )
=> ( ( Q @ I2 )
=> ( R2 @ I2 ) ) )
=> ( eventually @ A @ R2 @ F4 ) ) ) ) ).
% eventually_elim2
thf(fact_7946_eventually__conj,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A @ Q @ F4 )
=> ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ F4 ) ) ) ).
% eventually_conj
thf(fact_7947_eventually__True,axiom,
! [A: $tType,F4: filter @ A] :
( eventually @ A
@ ^ [X2: A] : $true
@ F4 ) ).
% eventually_True
thf(fact_7948_eventually__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F4 )
=> ( ( eventually @ A @ P @ F4 )
=> ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_mp
thf(fact_7949_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P: A > $o,C3: $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
| C3 )
@ F4 )
= ( ( eventually @ A @ P @ F4 )
| C3 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_7950_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C3: $o,P: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( C3
| ( P @ X2 ) )
@ F4 )
= ( C3
| ( eventually @ A @ P @ F4 ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_7951_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C3: $o,P: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( C3
=> ( P @ X2 ) )
@ F4 )
= ( C3
=> ( eventually @ A @ P @ F4 ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_7952_False__imp__not__eventually,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ! [X3: A] :
~ ( P @ X3 )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_7953_eventually__const__iff,axiom,
! [A: $tType,P: $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : P
@ F4 )
= ( P
| ( F4
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_7954_trivial__limit__def,axiom,
! [A: $tType,F4: filter @ A] :
( ( F4
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X2: A] : $false
@ F4 ) ) ).
% trivial_limit_def
thf(fact_7955_eventually__not__equal__at__infinity,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A] :
( eventually @ A
@ ^ [X2: A] : X2 != A2
@ ( at_infinity @ A ) ) ) ).
% eventually_not_equal_at_infinity
thf(fact_7956_eventually__at__infinity,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_infinity @ A ) )
= ( ? [B5: real] :
! [X2: A] :
( ( ord_less_eq @ real @ B5 @ ( real_V7770717601297561774m_norm @ A @ X2 ) )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_at_infinity
thf(fact_7957_filter__leD,axiom,
! [A: $tType,F4: filter @ A,F11: filter @ A,P: A > $o] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ F11 )
=> ( ( eventually @ A @ P @ F11 )
=> ( eventually @ A @ P @ F4 ) ) ) ).
% filter_leD
thf(fact_7958_filter__leI,axiom,
! [A: $tType,F11: filter @ A,F4: filter @ A] :
( ! [P8: A > $o] :
( ( eventually @ A @ P8 @ F11 )
=> ( eventually @ A @ P8 @ F4 ) )
=> ( ord_less_eq @ ( filter @ A ) @ F4 @ F11 ) ) ).
% filter_leI
thf(fact_7959_le__filter__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( filter @ A ) )
= ( ^ [F9: filter @ A,F10: filter @ A] :
! [P3: A > $o] :
( ( eventually @ A @ P3 @ F10 )
=> ( eventually @ A @ P3 @ F9 ) ) ) ) ).
% le_filter_def
thf(fact_7960_has__field__derivative__cong__ev,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,Y: A,S2: set @ A,F2: A > A,G: A > A,U2: A,V2: A,T2: set @ A] :
( ( X = Y )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S2 )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( ( U2 = V2 )
=> ( ( S2 = T2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( has_field_derivative @ A @ F2 @ U2 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
= ( has_field_derivative @ A @ G @ V2 @ ( topolo174197925503356063within @ A @ Y @ T2 ) ) ) ) ) ) ) ) ) ).
% has_field_derivative_cong_ev
thf(fact_7961_eventually__Lim__ident__at,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [P: A > A > $o,X: A,X5: set @ A] :
( ( eventually @ A
@ ( P
@ ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X @ X5 )
@ ^ [X2: A] : X2 ) )
@ ( topolo174197925503356063within @ A @ X @ X5 ) )
= ( eventually @ A @ ( P @ X ) @ ( topolo174197925503356063within @ A @ X @ X5 ) ) ) ) ).
% eventually_Lim_ident_at
thf(fact_7962_real__tendsto__sandwich,axiom,
! [B: $tType,F2: B > real,G: B > real,Net: filter @ B,H2: B > real,C2: real] :
( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ real @ ( G @ N3 ) @ ( H2 @ N3 ) )
@ Net )
=> ( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net )
=> ( ( filterlim @ B @ real @ H2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net )
=> ( filterlim @ B @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net ) ) ) ) ) ).
% real_tendsto_sandwich
thf(fact_7963_filterlim__at__within__not__equal,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ B )
=> ! [F2: A > B,A2: B,S: set @ B,F4: filter @ A,B2: B] :
( ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ A2 @ S ) @ F4 )
=> ( eventually @ A
@ ^ [W3: A] :
( ( member @ B @ ( F2 @ W3 ) @ S )
& ( ( F2 @ W3 )
!= B2 ) )
@ F4 ) ) ) ).
% filterlim_at_within_not_equal
thf(fact_7964_filterlim__at__infinity,axiom,
! [C: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: real,F2: C > A,F4: filter @ C] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ C @ A @ F2 @ ( at_infinity @ A ) @ F4 )
= ( ! [R5: real] :
( ( ord_less @ real @ C2 @ R5 )
=> ( eventually @ C
@ ^ [X2: C] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X2 ) ) )
@ F4 ) ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_7965_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P: A > $o,F4: filter @ A,F2: B > A,G6: filter @ B] :
( ( eventually @ A @ P @ F4 )
=> ( ( filterlim @ B @ A @ F2 @ F4 @ G6 )
=> ( eventually @ B
@ ^ [X2: B] : ( P @ ( F2 @ X2 ) )
@ G6 ) ) ) ).
% eventually_compose_filterlim
thf(fact_7966_filterlim__principal,axiom,
! [B: $tType,A: $tType,F2: A > B,S2: set @ B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( principal @ B @ S2 ) @ F4 )
= ( eventually @ A
@ ^ [X2: A] : ( member @ B @ ( F2 @ X2 ) @ S2 )
@ F4 ) ) ).
% filterlim_principal
thf(fact_7967_filterlim__cong,axiom,
! [A: $tType,B: $tType,F13: filter @ A,F14: filter @ A,F24: filter @ B,F25: filter @ B,F2: B > A,G: B > A] :
( ( F13 = F14 )
=> ( ( F24 = F25 )
=> ( ( eventually @ B
@ ^ [X2: B] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ F24 )
=> ( ( filterlim @ B @ A @ F2 @ F13 @ F24 )
= ( filterlim @ B @ A @ G @ F14 @ F25 ) ) ) ) ) ).
% filterlim_cong
thf(fact_7968_filterlim__iff,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F3: A > B,F26: filter @ B,F16: filter @ A] :
! [P3: B > $o] :
( ( eventually @ B @ P3 @ F26 )
=> ( eventually @ A
@ ^ [X2: A] : ( P3 @ ( F3 @ X2 ) )
@ F16 ) ) ) ) ).
% filterlim_iff
thf(fact_7969_tendsto__eventually,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,L: A,Net: filter @ B] :
( ( eventually @ B
@ ^ [X2: B] :
( ( F2 @ X2 )
= L )
@ Net )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ Net ) ) ) ).
% tendsto_eventually
thf(fact_7970_tendsto__imp__eventually__ne,axiom,
! [A: $tType,B: $tType] :
( ( topological_t1_space @ A )
=> ! [F2: B > A,C2: A,F4: filter @ B,C9: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( C2 != C9 )
=> ( eventually @ B
@ ^ [Z7: B] :
( ( F2 @ Z7 )
!= C9 )
@ F4 ) ) ) ) ).
% tendsto_imp_eventually_ne
thf(fact_7971_tendsto__discrete,axiom,
! [A: $tType,B: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ! [F2: B > A,Y: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
= ( eventually @ B
@ ^ [X2: B] :
( ( F2 @ X2 )
= Y )
@ F4 ) ) ) ).
% tendsto_discrete
thf(fact_7972_tendsto__cong,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,G: B > A,F4: filter @ B,C2: A] :
( ( eventually @ B
@ ^ [X2: B] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ F4 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
= ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 ) ) ) ) ).
% tendsto_cong
thf(fact_7973_filterlim__at__infinity__imp__eventually__ne,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F4 )
=> ( eventually @ A
@ ^ [Z7: A] :
( ( F2 @ Z7 )
!= C2 )
@ F4 ) ) ) ).
% filterlim_at_infinity_imp_eventually_ne
thf(fact_7974_Lim__transform__eventually,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ B )
=> ! [F2: A > B,L: B,F4: filter @ A,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ F4 )
=> ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ) ).
% Lim_transform_eventually
thf(fact_7975_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F2: A > B,F4: filter @ B,G6: filter @ A,F11: filter @ B,G7: filter @ A,F7: A > B] :
( ( filterlim @ A @ B @ F2 @ F4 @ G6 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F4 @ F11 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G7 @ G6 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
= ( F7 @ X2 ) )
@ G7 )
=> ( filterlim @ A @ B @ F7 @ F11 @ G7 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_7976_tendsto__sandwich,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,G: B > A,Net: filter @ B,H2: B > A,C2: A] :
( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ N3 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ ( G @ N3 ) @ ( H2 @ N3 ) )
@ Net )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( ( filterlim @ B @ A @ H2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net ) ) ) ) ) ) ).
% tendsto_sandwich
thf(fact_7977_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
=> ( ( ord_less @ A @ Y @ A2 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F2 @ X2 ) @ A2 )
@ F4 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_7978_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
=> ( ( ord_less @ A @ A2 @ Y )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ A2 @ ( F2 @ X2 ) )
@ F4 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_7979_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y: A,F2: B > A,F4: filter @ B] :
( ! [A6: A] :
( ( ord_less @ A @ A6 @ Y )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ A6 @ ( F2 @ X2 ) )
@ F4 ) )
=> ( ! [A6: A] :
( ( ord_less @ A @ Y @ A6 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F2 @ X2 ) @ A6 )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 ) ) ) ) ).
% order_tendstoI
thf(fact_7980_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,X: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
= ( ! [L2: A] :
( ( ord_less @ A @ L2 @ X )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ L2 @ ( F2 @ X2 ) )
@ F4 ) )
& ! [U: A] :
( ( ord_less @ A @ X @ U )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F2 @ X2 ) @ U )
@ F4 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_7981_filterlim__at,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,B2: A,S: set @ A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ B2 @ S ) @ F4 )
= ( ( eventually @ B
@ ^ [X2: B] :
( ( member @ A @ ( F2 @ X2 ) @ S )
& ( ( F2 @ X2 )
!= B2 ) )
@ F4 )
& ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 ) ) ) ) ).
% filterlim_at
thf(fact_7982_eventually__inf__principal,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,S: set @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F4 @ ( principal @ A @ S ) ) )
= ( eventually @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S )
=> ( P @ X2 ) )
@ F4 ) ) ).
% eventually_inf_principal
thf(fact_7983_le__principal,axiom,
! [A: $tType,F4: filter @ A,A3: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ ( principal @ A @ A3 ) )
= ( eventually @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 )
@ F4 ) ) ).
% le_principal
thf(fact_7984_eventually__at__left__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ X )
& ! [Y4: A] :
( ( ord_less @ A @ B5 @ Y4 )
=> ( ( ord_less @ A @ Y4 @ X )
=> ( P @ Y4 ) ) ) ) ) ) ) ).
% eventually_at_left_field
thf(fact_7985_eventually__at__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y: A,X: A,P: A > $o] :
( ( ord_less @ A @ Y @ X )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ X )
& ! [Y4: A] :
( ( ord_less @ A @ B5 @ Y4 )
=> ( ( ord_less @ A @ Y4 @ X )
=> ( P @ Y4 ) ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_7986_tendsto__def,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
= ( ! [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
=> ( ( member @ A @ L @ S6 )
=> ( eventually @ B
@ ^ [X2: B] : ( member @ A @ ( F2 @ X2 ) @ S6 )
@ F4 ) ) ) ) ) ) ).
% tendsto_def
thf(fact_7987_topological__tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,L: A,F4: filter @ B,S2: set @ A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( member @ A @ L @ S2 )
=> ( eventually @ B
@ ^ [X2: B] : ( member @ A @ ( F2 @ X2 ) @ S2 )
@ F4 ) ) ) ) ) ).
% topological_tendstoD
thf(fact_7988_topological__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [L: A,F2: B > A,F4: filter @ B] :
( ! [S4: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S4 )
=> ( ( member @ A @ L @ S4 )
=> ( eventually @ B
@ ^ [X2: B] : ( member @ A @ ( F2 @ X2 ) @ S4 )
@ F4 ) ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ).
% topological_tendstoI
thf(fact_7989_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: B > $o,F4: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F4 @ A3 ) ) )
= ( ? [Q8: A > B > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( eventually @ B @ ( Q8 @ X2 ) @ ( F4 @ X2 ) ) )
& ! [Y4: B] :
( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( Q8 @ X2 @ Y4 ) )
=> ( P @ Y4 ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_7990_eventually__at,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A,S2: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S2 ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X2: A] :
( ( member @ A @ X2 @ S2 )
=> ( ( ( X2 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A2 ) @ D5 ) )
=> ( P @ X2 ) ) ) ) ) ) ) ).
% eventually_at
thf(fact_7991_eventually__nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A] :
( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ A2 ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A2 ) @ D5 )
=> ( P @ X2 ) ) ) ) ) ) ).
% eventually_nhds_metric
thf(fact_7992_eventually__at__leftI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% eventually_at_leftI
thf(fact_7993_eventually__at__rightI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% eventually_at_rightI
thf(fact_7994_eventually__at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o,A2: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( eventually @ A
@ ^ [X2: A] : ( P @ ( plus_plus @ A @ X2 @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_7995_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L: A,F2: B > A,F4: filter @ B] :
( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ L @ ( F2 @ N3 ) )
@ F4 )
=> ( ! [X3: A] :
( ( ord_less @ A @ L @ X3 )
=> ( eventually @ B
@ ^ [N3: B] : ( ord_less @ A @ ( F2 @ N3 ) @ X3 )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% decreasing_tendsto
thf(fact_7996_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,L: A,F4: filter @ B] :
( ( eventually @ B
@ ^ [N3: B] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ L )
@ F4 )
=> ( ! [X3: A] :
( ( ord_less @ A @ X3 @ L )
=> ( eventually @ B
@ ^ [N3: B] : ( ord_less @ A @ X3 @ ( F2 @ N3 ) )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% increasing_tendsto
thf(fact_7997_filterlim__real__at__infinity__sequentially,axiom,
filterlim @ nat @ real @ ( semiring_1_of_nat @ real ) @ ( at_infinity @ real ) @ ( at_top @ nat ) ).
% filterlim_real_at_infinity_sequentially
thf(fact_7998_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F4 )
= ( ! [Z9: B] :
( ( ord_less @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X2 ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_7999_filterlim__atI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,C2: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] :
( ( F2 @ X2 )
!= C2 )
@ F4 )
=> ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ C2 @ ( top_top @ ( set @ A ) ) ) @ F4 ) ) ) ) ).
% filterlim_atI
thf(fact_8000_LIM__compose__eventually,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,B2: B,A2: A,G: B > C,C2: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
!= B2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_8001_tendsto__compose__eventually,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G: A > B,M: B,L: A,F2: C > A,F4: filter @ C] :
( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ M ) @ ( topolo174197925503356063within @ A @ L @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( eventually @ C
@ ^ [X2: C] :
( ( F2 @ X2 )
!= L )
@ F4 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( G @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ M )
@ F4 ) ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_8002_tendsto__of__nat,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( filterlim @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ).
% tendsto_of_nat
thf(fact_8003_isCont__cong,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,G: A > B,X: A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F2 )
= ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ G ) ) ) ) ).
% isCont_cong
thf(fact_8004_DERIV__cong__ev,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,Y: A,F2: A > A,G: A > A,U2: A,V2: A] :
( ( X = Y )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( ( U2 = V2 )
=> ( ( has_field_derivative @ A @ F2 @ U2 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A @ G @ V2 @ ( topolo174197925503356063within @ A @ Y @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% DERIV_cong_ev
thf(fact_8005_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F4 )
= ( ! [Z9: B] :
( ( ord_less @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ Z9 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_8006_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ ( F2 @ I4 ) @ A2 )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X @ A2 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_8007_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F2: B > A,X: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ A2 @ ( F2 @ I4 ) )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A2 @ X ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_8008_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F4: filter @ B,F2: B > A,X: A,G: B > A,Y: A] :
( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ A @ ( G @ X2 ) @ ( F2 @ X2 ) )
@ F4 )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ) ).
% tendsto_le
thf(fact_8009_metric__tendsto__imp__tendsto,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F2: C > A,A2: A,F4: filter @ C,G: C > B,B2: B] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( eventually @ C
@ ^ [X2: C] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G @ X2 ) @ B2 ) @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X2 ) @ A2 ) )
@ F4 )
=> ( filterlim @ C @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ F4 ) ) ) ) ).
% metric_tendsto_imp_tendsto
thf(fact_8010_eventually__floor__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] :
( ( archim6421214686448440834_floor @ B @ ( F2 @ X2 ) )
= ( archim6421214686448440834_floor @ B @ L ) )
@ F4 ) ) ) ) ).
% eventually_floor_eq
thf(fact_8011_eventually__ceiling__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] :
( ( archimedean_ceiling @ B @ ( F2 @ X2 ) )
= ( archimedean_ceiling @ B @ L ) )
@ F4 ) ) ) ) ).
% eventually_ceiling_eq
thf(fact_8012_eventually__at__right__to__0,axiom,
! [P: real > $o,A2: real] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
= ( eventually @ real
@ ^ [X2: real] : ( P @ ( plus_plus @ real @ X2 @ A2 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_right_to_0
thf(fact_8013_eventually__INF,axiom,
! [A: $tType,B: $tType,P: A > $o,F4: B > ( filter @ A ),B3: set @ B] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ B @ ( filter @ A ) @ F4 @ B3 ) ) )
= ( ? [X8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X8 @ B3 )
& ( finite_finite2 @ B @ X8 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ B @ ( filter @ A ) @ F4 @ X8 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_8014_tendsto__add__filterlim__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,C2: B,F4: filter @ A,G: A > B] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F4 )
=> ( ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity
thf(fact_8015_tendsto__add__filterlim__at__infinity_H,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A,G: A > B,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F4 )
=> ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_8016_eventually__at__left__to__right,axiom,
! [P: real > $o,A2: real] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
= ( eventually @ real
@ ^ [X2: real] : ( P @ ( uminus_uminus @ real @ X2 ) )
@ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A2 ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ A2 ) ) ) ) ) ).
% eventually_at_left_to_right
thf(fact_8017_continuous__arcosh__strong,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F2 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X2 ) )
@ F4 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X2: A] : ( arcosh @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_arcosh_strong
thf(fact_8018_eventually__at__right__real,axiom,
! [A2: real,B2: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( eventually @ real
@ ^ [X2: real] : ( member @ real @ X2 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ).
% eventually_at_right_real
thf(fact_8019_eventually__at__left__real,axiom,
! [B2: real,A2: real] :
( ( ord_less @ real @ B2 @ A2 )
=> ( eventually @ real
@ ^ [X2: real] : ( member @ real @ X2 @ ( set_or5935395276787703475ssThan @ real @ B2 @ A2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ).
% eventually_at_left_real
thf(fact_8020_eventually__at__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P: A > $o,A2: A,S2: set @ A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S2 ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X2: A] :
( ( member @ A @ X2 @ S2 )
=> ( ( ( X2 != A2 )
& ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A2 ) @ D5 ) )
=> ( P @ X2 ) ) ) ) ) ) ) ).
% eventually_at_le
thf(fact_8021_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C2: nat > A,K: nat,N: nat,B3: real] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( eventually @ A
@ ^ [Z7: A] :
( ord_less_eq @ real @ B3
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ Z7 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_8022_tendsto__compose__at,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,Y: B,F4: filter @ A,G: B > C,Z3: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ Y ) @ F4 )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ Z3 ) @ ( topolo174197925503356063within @ B @ Y @ ( top_top @ ( set @ B ) ) ) )
=> ( ( eventually @ A
@ ^ [W3: A] :
( ( ( F2 @ W3 )
= Y )
=> ( ( G @ Y )
= Z3 ) )
@ F4 )
=> ( filterlim @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ ( topolo7230453075368039082e_nhds @ C @ Z3 ) @ F4 ) ) ) ) ) ).
% tendsto_compose_at
thf(fact_8023_tendsto__imp__filterlim__at__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L6: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ L6 )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_lessThan @ B @ L6 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_8024_tendsto__imp__filterlim__at__right,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L6: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ L6 @ ( F2 @ X2 ) )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_greaterThan @ B @ L6 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_right
thf(fact_8025_tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F4: filter @ B,E2: real] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X2 ) @ L ) @ E2 )
@ F4 ) ) ) ) ).
% tendstoD
thf(fact_8026_tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F4: filter @ B] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X2 ) @ L ) @ E )
@ F4 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ).
% tendstoI
thf(fact_8027_tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F2: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
= ( ! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X2 ) @ L ) @ E3 )
@ F4 ) ) ) ) ) ).
% tendsto_iff
thf(fact_8028_eventually__Inf,axiom,
! [A: $tType,P: A > $o,B3: set @ ( filter @ A )] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B3 ) )
= ( ? [X8: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X8 @ B3 )
& ( finite_finite2 @ ( filter @ A ) @ X8 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ X8 ) ) ) ) ) ).
% eventually_Inf
thf(fact_8029_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_8030_eventually__at__right__to__top,axiom,
! [P: real > $o] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
= ( eventually @ real
@ ^ [X2: real] : ( P @ ( inverse_inverse @ real @ X2 ) )
@ ( at_top @ real ) ) ) ).
% eventually_at_right_to_top
thf(fact_8031_eventually__at__top__to__right,axiom,
! [P: real > $o] :
( ( eventually @ real @ P @ ( at_top @ real ) )
= ( eventually @ real
@ ^ [X2: real] : ( P @ ( inverse_inverse @ real @ X2 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_top_to_right
thf(fact_8032_tendsto__arcosh__strong,axiom,
! [B: $tType,F2: B > real,A2: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ A2 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X2 ) )
@ F4 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arcosh @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_arcosh_strong
thf(fact_8033_filterlim__at__top__at__left,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A2: A] :
( ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F2 @ ( G @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) )
=> ( ! [B6: A] :
( ( Q @ B6 )
=> ( ord_less @ A @ B6 @ A2 ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_left
thf(fact_8034_tendsto__mult__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,C2: A,F4: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A @ G @ ( at_infinity @ A ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_8035_tendsto__divide__0,axiom,
! [A: $tType,C: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: C > A,C2: A,F4: filter @ C,G: C > A] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( filterlim @ C @ A @ G @ ( at_infinity @ A ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_divide_0
thf(fact_8036_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F2: A > B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_8037_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B3: set @ A,F4: A > ( filter @ B ),P: B > $o] :
( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ B3 )
=> ! [B6: A] :
( ( member @ A @ B6 @ B3 )
=> ? [X6: A] :
( ( member @ A @ X6 @ B3 )
& ( ord_less_eq @ ( filter @ B ) @ ( F4 @ X6 ) @ ( inf_inf @ ( filter @ B ) @ ( F4 @ A6 ) @ ( F4 @ B6 ) ) ) ) ) )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F4 @ B3 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ B3 )
& ( eventually @ B @ P @ ( F4 @ X2 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_8038_filterlim__at__bot__at__right,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A2: A] :
( ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F2 @ ( G @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( ! [B6: A] :
( ( Q @ B6 )
=> ( ord_less @ A @ A2 @ B6 ) )
=> ( ( eventually @ B @ P @ ( at_bot @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_bot_at_right
thf(fact_8039_filterlim__at__infinity__conv__norm__at__top,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,G6: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ G6 )
= ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ ( at_top @ real )
@ G6 ) ) ) ).
% filterlim_at_infinity_conv_norm_at_top
thf(fact_8040_filterlim__norm__at__top__imp__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ ( at_top @ real )
@ F4 )
=> ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F4 ) ) ) ).
% filterlim_norm_at_top_imp_at_infinity
thf(fact_8041_filterlim__at__infinity__imp__norm__at__top,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_infinity_imp_norm_at_top
thf(fact_8042_tendsto__0__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F4: filter @ A,G: A > C,K6: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G @ X2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ K6 ) )
@ F4 )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F4 ) ) ) ) ).
% tendsto_0_le
thf(fact_8043_filterlim__at__withinI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,C2: A,F4: filter @ B,A3: set @ A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( member @ A @ ( F2 @ X2 ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ F4 )
=> ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ C2 @ A3 ) @ F4 ) ) ) ) ).
% filterlim_at_withinI
thf(fact_8044_tendsto__zero__powrI,axiom,
! [A: $tType,F2: A > real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ) ) ).
% tendsto_zero_powrI
thf(fact_8045_tendsto__powr2,axiom,
! [A: $tType,F2: A > real,A2: real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F4 ) ) ) ) ) ).
% tendsto_powr2
thf(fact_8046_tendsto__powr_H,axiom,
! [A: $tType,F2: A > real,A2: real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( ( A2
!= ( zero_zero @ real ) )
| ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F4 ) ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_powr'
thf(fact_8047_eventually__floor__less,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L ) ) @ ( F2 @ X2 ) )
@ F4 ) ) ) ) ).
% eventually_floor_less
thf(fact_8048_eventually__less__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L ) ) )
@ F4 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_8049_LIM__at__top__divide,axiom,
! [A: $tType,F2: A > real,A2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_8050_filterlim__at__top__iff__inverse__0,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F4 )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F4 )
= ( filterlim @ A @ real @ ( comp @ real @ real @ A @ ( inverse_inverse @ real ) @ F2 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% filterlim_at_top_iff_inverse_0
thf(fact_8051_filterlim__inverse__at__top__iff,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F4 )
=> ( ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F2 @ X2 ) )
@ ( at_top @ real )
@ F4 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% filterlim_inverse_at_top_iff
thf(fact_8052_filterlim__inverse__at__top,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F2 @ X2 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_inverse_at_top
thf(fact_8053_filterlim__inverse__at__bot,axiom,
! [A: $tType,F2: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( F2 @ X2 ) @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( inverse_inverse @ real @ ( F2 @ X2 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ).
% filterlim_inverse_at_bot
thf(fact_8054_lhopital__at__top__at__top,axiom,
! [F2: real > real,A2: real,G: real > real,F7: real > real,G4: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_top
thf(fact_8055_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_8056_lim__infinity__imp__sequentially,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: real > A,L: A] :
( ( filterlim @ real @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( semiring_1_of_nat @ real @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% lim_infinity_imp_sequentially
thf(fact_8057_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G: A > B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( inverse_inverse @ B @ ( G @ X2 ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F4 )
= ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F4 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_8058_lhopital,axiom,
! [F2: real > real,X: real,G: real > real,G4: real > real,F7: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital
thf(fact_8059_lhopital__right__at__top__at__top,axiom,
! [F2: real > real,A2: real,G: real > real,F7: real > real,G4: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_top
thf(fact_8060_lhopital__at__top__at__bot,axiom,
! [F2: real > real,A2: real,G: real > real,F7: real > real,G4: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_bot
thf(fact_8061_lhopital__left__at__top__at__top,axiom,
! [F2: real > real,A2: real,G: real > real,F7: real > real,G4: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_top
thf(fact_8062_filterlim__divide__at__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,C2: A,F4: filter @ A,G: A > A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( filterlim @ A @ A @ G @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F4 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_8063_lhospital__at__top__at__top,axiom,
! [G: real > real,G4: real > real,F2: real > real,F7: real > real,X: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ real ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ real ) ) ) ) ) ) ) ).
% lhospital_at_top_at_top
thf(fact_8064_lhopital__at__top,axiom,
! [G: real > real,X: real,G4: real > real,F2: real > real,F7: real > real,Y: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top
thf(fact_8065_lhopital__right__0,axiom,
! [F0: real > real,G0: real > real,G4: real > real,F7: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F0 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G0 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G0 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F0 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G0 @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F0 @ X2 ) @ ( G0 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0
thf(fact_8066_lhopital__right,axiom,
! [F2: real > real,X: real,G: real > real,G4: real > real,F7: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right
thf(fact_8067_filterlim__realpow__sequentially__gt1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ) ).
% filterlim_realpow_sequentially_gt1
thf(fact_8068_lhopital__left,axiom,
! [F2: real > real,X: real,G: real > real,G4: real > real,F7: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_left
thf(fact_8069_lhopital__right__at__top__at__bot,axiom,
! [F2: real > real,A2: real,G: real > real,F7: real > real,G4: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_bot
thf(fact_8070_lhopital__left__at__top__at__bot,axiom,
! [F2: real > real,A2: real,G: real > real,F7: real > real,G4: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_bot
thf(fact_8071_lim__at__infinity__0,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) )
= ( filterlim @ A @ A @ ( comp @ A @ A @ A @ F2 @ ( inverse_inverse @ A ) ) @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% lim_at_infinity_0
thf(fact_8072_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A] :
( ( filterlim @ A @ A
@ ^ [X2: A] : ( F2 @ ( divide_divide @ A @ ( one_one @ A ) @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_8073_lhopital__right__0__at__top,axiom,
! [G: real > real,G4: real > real,F2: real > real,F7: real > real,X: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0_at_top
thf(fact_8074_lhopital__right__at__top,axiom,
! [G: real > real,X: real,G4: real > real,F2: real > real,F7: real > real,Y: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G4 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F7 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G4 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F7 @ X2 ) @ ( G4 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top
thf(fact_8075_Bfun__metric__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F3: A > B,F9: filter @ A] :
? [Y4: B,K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X2 ) @ Y4 ) @ K7 )
@ F9 ) ) ) ) ) ).
% Bfun_metric_def
thf(fact_8076_BfunE,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F4: filter @ A] :
( ( bfun @ A @ B @ F2 @ F4 )
=> ~ ! [B7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B7 )
=> ~ ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ B7 )
@ F4 ) ) ) ) ).
% BfunE
thf(fact_8077_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_8078_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat
@ ^ [N3: nat] : ( P @ ( plus_plus @ nat @ N3 @ K ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_8079_eventually__False__sequentially,axiom,
~ ( eventually @ nat
@ ^ [N3: nat] : $false
@ ( at_top @ nat ) ) ).
% eventually_False_sequentially
thf(fact_8080_sequentially__offset,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [I4: nat] : ( P @ ( plus_plus @ nat @ I4 @ K ) )
@ ( at_top @ nat ) ) ) ).
% sequentially_offset
thf(fact_8081_summable__cong,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( eventually @ nat
@ ^ [X2: nat] :
( ( F2 @ X2 )
= ( G @ X2 ) )
@ ( at_top @ nat ) )
=> ( ( summable @ A @ F2 )
= ( summable @ A @ G ) ) ) ) ).
% summable_cong
thf(fact_8082_Bseq__eventually__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: nat > A,G: nat > B] :
( ( eventually @ nat
@ ^ [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( real_V7770717601297561774m_norm @ B @ ( G @ N3 ) ) )
@ ( at_top @ nat ) )
=> ( ( bfun @ nat @ B @ G @ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_eventually_mono
thf(fact_8083_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( P @ N3 ) ) ) ) ).
% eventually_sequentially
thf(fact_8084_eventually__sequentiallyI,axiom,
! [C2: nat,P: nat > $o] :
( ! [X3: nat] :
( ( ord_less_eq @ nat @ C2 @ X3 )
=> ( P @ X3 ) )
=> ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_8085_le__sequentially,axiom,
! [F4: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F4 @ ( at_top @ nat ) )
= ( ! [N7: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N7 ) @ F4 ) ) ) ).
% le_sequentially
thf(fact_8086_countable__basis__at__decseq,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
~ ! [A10: nat > ( set @ A )] :
( ! [I3: nat] : ( topolo1002775350975398744n_open @ A @ ( A10 @ I3 ) )
=> ( ! [I3: nat] : ( member @ A @ X @ ( A10 @ I3 ) )
=> ~ ! [S11: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S11 )
=> ( ( member @ A @ X @ S11 )
=> ( eventually @ nat
@ ^ [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( A10 @ I4 ) @ S11 )
@ ( at_top @ nat ) ) ) ) ) ) ) ).
% countable_basis_at_decseq
thf(fact_8087_Bfun__const,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ! [C2: B,F4: filter @ A] :
( bfun @ A @ B
@ ^ [Uu: A] : C2
@ F4 ) ) ).
% Bfun_const
thf(fact_8088_Bseq__minus__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A
@ ^ [N3: nat] : ( uminus_uminus @ A @ ( X5 @ N3 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) ) ) ) ).
% Bseq_minus_iff
thf(fact_8089_Bseq__subseq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > nat] :
( ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_subseq
thf(fact_8090_Bseq__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( bfun @ nat @ A
@ ^ [N3: nat] : ( F2 @ ( suc @ N3 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ).
% Bseq_Suc_iff
thf(fact_8091_Bseq__offset,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,K: nat] :
( ( bfun @ nat @ A
@ ^ [N3: nat] : ( X5 @ ( plus_plus @ nat @ N3 @ K ) )
@ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) ) ) ) ).
% Bseq_offset
thf(fact_8092_Bseq__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X5: nat > A,K: nat] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [N3: nat] : ( X5 @ ( plus_plus @ nat @ N3 @ K ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_ignore_initial_segment
thf(fact_8093_Bseq__add,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X2: nat] : ( plus_plus @ A @ ( F2 @ X2 ) @ C2 )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_add
thf(fact_8094_Bseq__add__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( bfun @ nat @ A
@ ^ [X2: nat] : ( plus_plus @ A @ ( F2 @ X2 ) @ C2 )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ).
% Bseq_add_iff
thf(fact_8095_Bseq__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) )
=> ( ( bfun @ nat @ A @ G @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X2: nat] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ nat ) ) ) ) ) ).
% Bseq_mult
thf(fact_8096_BseqI_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A,K6: real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X5 @ N2 ) ) @ K6 )
=> ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) ) ) ) ).
% BseqI'
thf(fact_8097_filterlim__int__sequentially,axiom,
filterlim @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( at_top @ int ) @ ( at_top @ nat ) ).
% filterlim_int_sequentially
thf(fact_8098_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X2: nat] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_8099_filterlim__int__of__nat__at__topD,axiom,
! [A: $tType,F2: int > A,F4: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X2: nat] : ( F2 @ ( semiring_1_of_nat @ int @ X2 ) )
@ F4
@ ( at_top @ nat ) )
=> ( filterlim @ int @ A @ F2 @ F4 @ ( at_top @ int ) ) ) ).
% filterlim_int_of_nat_at_topD
thf(fact_8100_Bseq__eq__bounded,axiom,
! [F2: nat > real,A2: real,B2: real] :
( ( ord_less_eq @ ( set @ real ) @ ( image @ nat @ real @ F2 @ ( top_top @ ( set @ nat ) ) ) @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( bfun @ nat @ real @ F2 @ ( at_top @ nat ) ) ) ).
% Bseq_eq_bounded
thf(fact_8101_summable__comparison__test__ev,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) )
@ ( at_top @ nat ) )
=> ( ( summable @ real @ G )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_comparison_test_ev
thf(fact_8102_BseqD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
=> ? [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
& ! [N9: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X5 @ N9 ) ) @ K9 ) ) ) ) ).
% BseqD
thf(fact_8103_BseqE,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
=> ~ ! [K9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K9 )
=> ~ ! [N9: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X5 @ N9 ) ) @ K9 ) ) ) ) ).
% BseqE
thf(fact_8104_BseqI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [K6: real,X5: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X5 @ N2 ) ) @ K6 )
=> ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) ) ) ) ) ).
% BseqI
thf(fact_8105_Bseq__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
= ( ? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X5 @ N3 ) ) @ K7 ) ) ) ) ) ).
% Bseq_def
thf(fact_8106_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
= ( ? [N7: nat] :
! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X5 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N7 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_8107_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
= ( ? [N7: nat] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X5 @ N3 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N7 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_8108_Bseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( bfun @ nat @ real @ ( power_power @ real @ X ) @ ( at_top @ nat ) ) ) ) ).
% Bseq_realpow
thf(fact_8109_BfunI,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,K6: real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ K6 )
@ F4 )
=> ( bfun @ A @ B @ F2 @ F4 ) ) ) ).
% BfunI
thf(fact_8110_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [X2: A] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X5 @ N3 ) @ ( uminus_uminus @ A @ X2 ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_8111_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X5: nat > A] :
( ( bfun @ nat @ A @ X5 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [N7: nat] :
! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X5 @ N3 ) @ ( uminus_uminus @ A @ ( X5 @ N7 ) ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_8112_Bfun__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A2: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( bfun @ B @ A
@ ^ [X2: B] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ F4 ) ) ) ) ).
% Bfun_inverse
thf(fact_8113_Bfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F3: A > B,F9: filter @ A] :
? [K7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K7 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) @ K7 )
@ F9 ) ) ) ) ) ).
% Bfun_def
thf(fact_8114_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [M3: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M3 @ N3 ) ) ) @ ( G @ M3 ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_Cauchy'
thf(fact_8115_sequentially__imp__eventually__at__right,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ! [F5: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ A2 @ ( F5 @ N9 ) )
=> ( ! [N9: nat] : ( ord_less @ A @ ( F5 @ N9 ) @ B2 )
=> ( ( order_antimono @ nat @ A @ F5 )
=> ( ( filterlim @ nat @ A @ F5 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N3: nat] : ( P @ ( F5 @ N3 ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_right
thf(fact_8116_decseq__const,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [K: A] :
( order_antimono @ nat @ A
@ ^ [X2: nat] : K ) ) ).
% decseq_const
thf(fact_8117_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X2: A] :
! [Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( P @ Y4 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_8118_eventually__all__finite,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [P: A > B > $o,Net: filter @ A] :
( ! [Y3: B] :
( eventually @ A
@ ^ [X2: A] : ( P @ X2 @ Y3 )
@ Net )
=> ( eventually @ A
@ ^ [X2: A] :
! [X8: B] : ( P @ X2 @ X8 )
@ Net ) ) ) ).
% eventually_all_finite
thf(fact_8119_decseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,I: nat,J: nat] :
( ( order_antimono @ nat @ A @ F2 )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( F2 @ J ) @ ( F2 @ I ) ) ) ) ) ).
% decseqD
thf(fact_8120_decseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X8: nat > A] :
! [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ M3 ) ) ) ) ) ) ).
% decseq_def
thf(fact_8121_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ).
% antimonoD
thf(fact_8122_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ).
% antimonoE
thf(fact_8123_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) )
=> ( order_antimono @ A @ B @ F2 ) ) ) ).
% antimonoI
thf(fact_8124_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F3: A > B] :
! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ Y4 ) @ ( F3 @ X2 ) ) ) ) ) ) ).
% antimono_def
thf(fact_8125_decseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F3: nat > A] :
! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N3 ) ) @ ( F3 @ N3 ) ) ) ) ) ).
% decseq_Suc_iff
thf(fact_8126_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X5: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X5 @ ( suc @ N2 ) ) @ ( X5 @ N2 ) )
=> ( order_antimono @ nat @ A @ X5 ) ) ) ).
% decseq_SucI
thf(fact_8127_decseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: nat > A,I: nat] :
( ( order_antimono @ nat @ A @ A3 )
=> ( ord_less_eq @ A @ ( A3 @ ( suc @ I ) ) @ ( A3 @ I ) ) ) ) ).
% decseq_SucD
thf(fact_8128_decseq__bounded,axiom,
! [X5: nat > real,B3: real] :
( ( order_antimono @ nat @ real @ X5 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ B3 @ ( X5 @ I2 ) )
=> ( bfun @ nat @ real @ X5 @ ( at_top @ nat ) ) ) ) ).
% decseq_bounded
thf(fact_8129_Collect__all__eq,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ( collect @ A
@ ^ [X2: A] :
! [X8: B] : ( P @ X2 @ X8 ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y4: B] :
( collect @ A
@ ^ [X2: A] : ( P @ X2 @ Y4 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_all_eq
thf(fact_8130_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,D4: B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F3: A > B] :
! [X2: A] :
( ( ( member @ A @ X2 @ A3 )
=> ( member @ B @ ( F3 @ X2 ) @ B3 ) )
& ( ~ ( member @ A @ X2 @ A3 )
=> ( ( F3 @ X2 )
= D4 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_8131_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ! [Y4: A] :
( ( P3 @ Y4 )
=> ( ord_less_eq @ A @ X2 @ Y4 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_8132_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ! [Y4: A] :
( ( P3 @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_8133_decseq__ge,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X5: nat > A,L6: A,N: nat] :
( ( order_antimono @ nat @ A @ X5 )
=> ( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ L6 @ ( X5 @ N ) ) ) ) ) ).
% decseq_ge
thf(fact_8134_decseq__convergent,axiom,
! [X5: nat > real,B3: real] :
( ( order_antimono @ nat @ real @ X5 )
=> ( ! [I2: nat] : ( ord_less_eq @ real @ B3 @ ( X5 @ I2 ) )
=> ~ ! [L7: real] :
( ( filterlim @ nat @ real @ X5 @ ( topolo7230453075368039082e_nhds @ real @ L7 ) @ ( at_top @ nat ) )
=> ~ ! [I3: nat] : ( ord_less_eq @ real @ L7 @ ( X5 @ I3 ) ) ) ) ) ).
% decseq_convergent
thf(fact_8135_nhds__countable,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
~ ! [X10: nat > ( set @ A )] :
( ( order_antimono @ nat @ ( set @ A ) @ X10 )
=> ( ! [N9: nat] : ( topolo1002775350975398744n_open @ A @ ( X10 @ N9 ) )
=> ( ! [N9: nat] : ( member @ A @ X @ ( X10 @ N9 ) )
=> ( ( topolo7230453075368039082e_nhds @ A @ X )
!= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ nat @ ( filter @ A )
@ ^ [N3: nat] : ( principal @ A @ ( X10 @ N3 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ) ).
% nhds_countable
thf(fact_8136_LIMSEQ__INF,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X5: nat > A] :
( ( order_antimono @ nat @ A @ X5 )
=> ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ ( complete_Inf_Inf @ A @ ( image @ nat @ A @ X5 @ ( top_top @ ( set @ nat ) ) ) ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_INF
thf(fact_8137_INF__Lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X5: nat > A,L: A] :
( ( order_antimono @ nat @ A @ X5 )
=> ( ( filterlim @ nat @ A @ X5 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ nat @ A @ X5 @ ( top_top @ ( set @ nat ) ) ) )
= L ) ) ) ) ).
% INF_Lim
thf(fact_8138_tendsto__at__right__sequentially,axiom,
! [C: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: B,B2: B,X5: B > C,L6: C] :
( ( ord_less @ B @ A2 @ B2 )
=> ( ! [S4: nat > B] :
( ! [N9: nat] : ( ord_less @ B @ A2 @ ( S4 @ N9 ) )
=> ( ! [N9: nat] : ( ord_less @ B @ ( S4 @ N9 ) @ B2 )
=> ( ( order_antimono @ nat @ B @ S4 )
=> ( ( filterlim @ nat @ B @ S4 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C
@ ^ [N3: nat] : ( X5 @ ( S4 @ N3 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L6 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ C @ X5 @ ( topolo7230453075368039082e_nhds @ C @ L6 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_greaterThan @ B @ A2 ) ) ) ) ) ) ).
% tendsto_at_right_sequentially
thf(fact_8139_summable__bounded__partials,axiom,
! [A: $tType] :
( ( ( real_V8037385150606011577_space @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [X02: nat] :
! [A5: nat] :
( ( ord_less_eq @ nat @ X02 @ A5 )
=> ! [B5: nat] :
( ( ord_less @ nat @ A5 @ B5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or3652927894154168847AtMost @ nat @ A5 @ B5 ) ) ) @ ( G @ A5 ) ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_bounded_partials
thf(fact_8140_max__ext__eq,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X8: set @ A,Y8: set @ A] :
( ( finite_finite2 @ A @ X8 )
& ( finite_finite2 @ A @ Y8 )
& ( Y8
!= ( bot_bot @ ( set @ A ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ Y8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R6 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_8141_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U2: A] :
( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U2 ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U2 ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_8142_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanAtMost_empty
thf(fact_8143_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_8144_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K @ L ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_8145_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioc_iff
thf(fact_8146_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_8147_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanAtMost
thf(fact_8148_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanAtMost
thf(fact_8149_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_8150_ivl__disj__un__two_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U2 ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_8151_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A2 @ B2 )
= ( set_or3652927894154168847AtMost @ A @ C2 @ D4 ) )
= ( ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ D4 @ C2 ) )
| ( ( A2 = C2 )
& ( B2 = D4 ) ) ) ) ) ).
% Ioc_inj
thf(fact_8152_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D4 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D4 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_8153_ivl__disj__un__two_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U2 ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_8154_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D4 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ord_less_eq @ A @ D4 @ C2 )
| ( ord_less_eq @ A @ B2 @ C2 )
| ( ord_less_eq @ A @ D4 @ A2 ) ) ) ) ).
% Ioc_disjoint
thf(fact_8155_ivl__disj__un__one_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U2 ) )
= ( set_ord_atMost @ A @ U2 ) ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_8156_open__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( ord_less @ A @ Y @ X )
=> ? [B6: A] :
( ( ord_less @ A @ B6 @ X )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B6 @ X ) @ S2 ) ) ) ) ) ) ).
% open_left
thf(fact_8157_ivl__disj__un__one_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U2 ) @ ( set_ord_greaterThan @ A @ U2 ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(5)
thf(fact_8158_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N3: nat,M3: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ ( suc @ M3 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_8159_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioc
thf(fact_8160_sum_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.head
thf(fact_8161_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.head
thf(fact_8162_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D4 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D4 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_8163_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D4 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D4 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_8164_ivl__disj__un__two__touch_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U2 ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_8165_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D4 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D4 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_8166_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A5: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B5 ) @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_8167_ivl__disj__un__two_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U2 ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_8168_ivl__disj__un__two__touch_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U2 ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_8169_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ ( suc @ I ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_8170_ivl__disj__un__singleton_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less_eq @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U2 ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U2 ) ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_8171_ivl__disj__un__two_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U2: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U2 ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U2 ) ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_8172_ivl__disj__un__singleton_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U2: A] :
( ( ord_less @ A @ L @ U2 )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U2 ) @ ( insert @ A @ U2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U2 ) ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_8173_max__extp__max__ext__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X2: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R2 ) )
= ( ^ [X2: set @ A,Y4: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X2 @ Y4 ) @ ( max_ext @ A @ R2 ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_8174_max__extp__eq,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R5: A > A > $o,X2: set @ A,Y4: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X2 @ Y4 ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_8175_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
@ ^ [X2: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R6 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_8176_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ I ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_8177_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S2: set @ A] :
( ! [A6: A,B6: A,X3: A] :
( ( member @ A @ A6 @ S2 )
=> ( ( member @ A @ B6 @ S2 )
=> ( ( ord_less_eq @ A @ A6 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B6 )
=> ( member @ A @ X3 @ S2 ) ) ) ) )
=> ? [A6: A,B6: A] :
( ( S2
= ( bot_bot @ ( set @ A ) ) )
| ( S2
= ( top_top @ ( set @ A ) ) )
| ( S2
= ( set_ord_lessThan @ A @ B6 ) )
| ( S2
= ( set_ord_atMost @ A @ B6 ) )
| ( S2
= ( set_ord_greaterThan @ A @ A6 ) )
| ( S2
= ( set_ord_atLeast @ A @ A6 ) )
| ( S2
= ( set_or5935395276787703475ssThan @ A @ A6 @ B6 ) )
| ( S2
= ( set_or3652927894154168847AtMost @ A @ A6 @ B6 ) )
| ( S2
= ( set_or7035219750837199246ssThan @ A @ A6 @ B6 ) )
| ( S2
= ( set_or1337092689740270186AtMost @ A @ A6 @ B6 ) ) ) ) ) ).
% interval_cases
thf(fact_8178_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G: B > A,Y6: set @ B,X5: set @ A,F4: filter @ B,F2: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ Y6 ) @ X5 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( member @ B @ X2 @ Y6 )
@ F4 )
=> ( ( map_filter_on @ A @ C @ X5 @ F2 @ ( map_filter_on @ B @ A @ Y6 @ G @ F4 ) )
= ( map_filter_on @ B @ C @ Y6 @ ( comp @ A @ C @ B @ F2 @ G ) @ F4 ) ) ) ) ).
% map_filter_on_comp
thf(fact_8179_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_8180_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
= ( ord_less_eq @ A @ K @ I ) ) ) ).
% atLeast_iff
% Type constructors (779)
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A17: $tType,A18: $tType] :
( ( comple6319245703460814977attice @ A18 )
=> ( condit1219197933456340205attice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
! [A17: $tType,A18: $tType] :
( ( counta3822494911875563373attice @ A18 )
=> ( counta3822494911875563373attice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A17: $tType,A18: $tType] :
( ( comple592849572758109894attice @ A18 )
=> ( comple592849572758109894attice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
! [A17: $tType,A18: $tType] :
( ( comple489889107523837845lgebra @ A18 )
=> ( comple489889107523837845lgebra @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A17: $tType,A18: $tType] :
( ( comple6319245703460814977attice @ A18 )
=> ( comple6319245703460814977attice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A17: $tType,A18: $tType] :
( ( boolea8198339166811842893lgebra @ A18 )
=> ( boolea8198339166811842893lgebra @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A17: $tType,A18: $tType] :
( ( comple6319245703460814977attice @ A18 )
=> ( comple9053668089753744459l_ccpo @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A17: $tType,A18: $tType] :
( ( semilattice_sup @ A18 )
=> ( semilattice_sup @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A17: $tType,A18: $tType] :
( ( semilattice_inf @ A18 )
=> ( semilattice_inf @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A17: $tType,A18: $tType] :
( ( complete_Sup @ A18 )
=> ( complete_Sup @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OInf,axiom,
! [A17: $tType,A18: $tType] :
( ( complete_Inf @ A18 )
=> ( complete_Inf @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A17: $tType,A18: $tType] :
( ( order_top @ A18 )
=> ( order_top @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A17: $tType,A18: $tType] :
( ( order_bot @ A18 )
=> ( order_bot @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A17: $tType,A18: $tType] :
( ( preorder @ A18 )
=> ( preorder @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A17: $tType,A18: $tType] :
( ( ( finite_finite @ A17 )
& ( finite_finite @ A18 ) )
=> ( finite_finite @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A17: $tType,A18: $tType] :
( ( lattice @ A18 )
=> ( lattice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A17: $tType,A18: $tType] :
( ( order @ A18 )
=> ( order @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A17: $tType,A18: $tType] :
( ( top @ A18 )
=> ( top @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A17: $tType,A18: $tType] :
( ( ord @ A18 )
=> ( ord @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A17: $tType,A18: $tType] :
( ( bot @ A18 )
=> ( bot @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A17: $tType,A18: $tType] :
( ( uminus @ A18 )
=> ( uminus @ ( A17 > A18 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,axiom,
condit1219197933456340205attice @ int ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
euclid8789492081693882211th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
topolo4958980785337419405_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
topolo1944317154257567458pology @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Odiscrete__topology,axiom,
topolo8865339358273720382pology @ int ).
thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__mult,axiom,
topolo4987421752381908075d_mult @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
topolo5987344860129210374id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
topolo2564578578187576103pology @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri2026040879449505780visors @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
topolo4211221413907600880p_mult @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
euclid5891614535332579305n_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
semiri6843258321239162965malize @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__mult,axiom,
topolo1898628316856586783d_mult @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
topolo6943815403480290642id_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s4317794764714335236cancel @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
topological_t2_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot1__space,axiom,
topological_t1_space @ int ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_2,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_3,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___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___Complete__Lattices_OSup_4,axiom,
complete_Sup @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OInf_5,axiom,
complete_Inf @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ int ).
thf(tcon_Int_Oint___Parity_Oring__parity,axiom,
ring_parity @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_6,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
idom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Oidom__divide,axiom,
idom_divide @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0 @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Lattices_Olattice_7,axiom,
lattice @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
semiring_gcd @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
semiring_Gcd @ int ).
thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
mult_zero @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_8,axiom,
order @ int ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int ).
thf(tcon_Int_Oint___Rings_Osemidom,axiom,
semidom @ int ).
thf(tcon_Int_Oint___Orderings_Oord_9,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_10,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
ring_gcd @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_11,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_12,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_13,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_14,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_15,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_16,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_17,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_18,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_19,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_20,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_21,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_22,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_23,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_24,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_25,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_26,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_27,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_28,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_29,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_30,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_31,axiom,
topolo8865339358273720382pology @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_32,axiom,
topolo4987421752381908075d_mult @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_33,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_34,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_35,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_36,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_37,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_38,axiom,
topolo4211221413907600880p_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_39,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_40,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_41,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_42,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__mult_43,axiom,
topolo1898628316856586783d_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_44,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_45,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_46,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_47,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_48,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_49,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_50,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space_51,axiom,
topological_t1_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_52,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_53,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_54,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_55,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_56,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_57,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_58,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_59,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_60,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_61,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_62,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_63,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_64,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_65,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_66,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_67,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_68,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_69,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_70,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OSup_71,axiom,
complete_Sup @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OInf_72,axiom,
complete_Inf @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_73,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_74,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_75,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_76,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_77,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_78,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_79,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_80,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_81,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_82,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_83,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_84,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_85,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_86,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_87,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_88,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_89,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_90,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_91,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_92,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_93,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_94,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom_95,axiom,
semidom @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_96,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_97,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Power_Opower_98,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_99,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_100,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_101,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_102,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_103,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_104,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_105,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_106,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_107,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_108,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_109,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_110,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_111,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_112,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_113,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_114,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_115,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_116,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_117,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_118,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_119,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_120,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_121,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_122,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_123,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_124,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_125,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_126,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_127,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_128,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_129,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_130,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_131,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_132,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_133,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_134,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_135,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_136,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_137,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_138,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_139,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_140,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_141,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_142,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_143,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_144,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_145,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_146,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_147,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_148,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_149,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_150,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_151,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_152,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_153,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_154,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_155,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_156,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_157,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_158,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_159,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_160,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_161,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__neq__one_162,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_163,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_164,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_165,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_166,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_167,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_168,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_169,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_170,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_171,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_172,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_173,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_174,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_175,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_176,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_177,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_178,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_179,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_180,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_181,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_182,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_183,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_184,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_185,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_186,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_187,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_188,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_189,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_190,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_191,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_192,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_193,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_194,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_195,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_196,axiom,
! [A17: $tType] : ( condit1219197933456340205attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_197,axiom,
! [A17: $tType] : ( counta3822494911875563373attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_198,axiom,
! [A17: $tType] : ( comple592849572758109894attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_199,axiom,
! [A17: $tType] : ( comple489889107523837845lgebra @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_200,axiom,
! [A17: $tType] : ( comple6319245703460814977attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_201,axiom,
! [A17: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_202,axiom,
! [A17: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_203,axiom,
! [A17: $tType] : ( semilattice_sup @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_204,axiom,
! [A17: $tType] : ( semilattice_inf @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_205,axiom,
! [A17: $tType] : ( complete_Sup @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OInf_206,axiom,
! [A17: $tType] : ( complete_Inf @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_207,axiom,
! [A17: $tType] : ( order_top @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_208,axiom,
! [A17: $tType] : ( order_bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_209,axiom,
! [A17: $tType] : ( preorder @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_210,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( finite_finite @ ( set @ A17 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_211,axiom,
! [A17: $tType] : ( lattice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_212,axiom,
! [A17: $tType] : ( order @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_213,axiom,
! [A17: $tType] : ( top @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_214,axiom,
! [A17: $tType] : ( ord @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_215,axiom,
! [A17: $tType] : ( bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_216,axiom,
! [A17: $tType] : ( uminus @ ( set @ A17 ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_217,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_218,axiom,
counta3822494911875563373attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_219,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_220,axiom,
comple489889107523837845lgebra @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_221,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_222,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_223,axiom,
topolo8865339358273720382pology @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_224,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_225,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_226,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_227,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_228,axiom,
topological_t1_space @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_229,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_230,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_231,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_232,axiom,
complete_Sup @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OInf_233,axiom,
complete_Inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_234,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_235,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_236,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_237,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_238,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_239,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_240,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_241,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_242,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_243,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_244,axiom,
uminus @ $o ).
thf(tcon_List_Olist___Nat_Osize_245,axiom,
! [A17: $tType] : ( size @ ( list @ A17 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_246,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_247,axiom,
condit1219197933456340205attice @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_248,axiom,
semiri1453513574482234551roduct @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
condit5016429287641298734tinuum @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_249,axiom,
ordere1937475149494474687imp_le @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinear__continuum__topology,axiom,
topolo8458572112393995274pology @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology,axiom,
topolo3112930676232923870pology @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra,axiom,
real_V8999393235501362500lgebra @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom,
real_V2822296259951069270ebra_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_250,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_251,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_252,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_253,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_254,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_255,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_256,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_257,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_258,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_259,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_260,axiom,
linord715952674999750819strict @ real ).
thf(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_261,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_262,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_263,axiom,
linord4140545234300271783up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_264,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_265,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_266,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_267,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_268,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_269,axiom,
semiri3467727345109120633visors @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
real_V7819770556892013058_space @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
topolo1287966508704411220up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_270,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_271,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_272,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_273,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_274,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_275,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_276,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_277,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_278,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_279,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_280,axiom,
comm_s4317794764714335236cancel @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
topolo1633459387980952147up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_281,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot1__space_282,axiom,
topological_t1_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___Complete__Lattices_OSup_307,axiom,
complete_Sup @ real ).
thf(tcon_Real_Oreal___Complete__Lattices_OInf_308,axiom,
complete_Inf @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_309,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_310,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_311,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_312,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_313,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_314,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_315,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_316,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_317,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_318,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_319,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_320,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_321,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_322,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_323,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide_324,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_325,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_326,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_327,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_328,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_329,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_330,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_331,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_332,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_333,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_334,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_335,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_336,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_337,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_338,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_339,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom_340,axiom,
semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_341,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_342,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_343,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_344,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_345,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_346,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_347,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_348,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_349,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_350,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_351,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_352,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_353,axiom,
dvd @ real ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_354,axiom,
finite_finite @ char ).
thf(tcon_String_Ochar___Nat_Osize_355,axiom,
size @ char ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_356,axiom,
! [A17: $tType] : ( condit1219197933456340205attice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_357,axiom,
! [A17: $tType] : ( counta3822494911875563373attice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_358,axiom,
! [A17: $tType] : ( comple6319245703460814977attice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_359,axiom,
! [A17: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_360,axiom,
! [A17: $tType] : ( semilattice_sup @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_361,axiom,
! [A17: $tType] : ( semilattice_inf @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OSup_362,axiom,
! [A17: $tType] : ( complete_Sup @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OInf_363,axiom,
! [A17: $tType] : ( complete_Inf @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_364,axiom,
! [A17: $tType] : ( order_top @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_365,axiom,
! [A17: $tType] : ( order_bot @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_366,axiom,
! [A17: $tType] : ( preorder @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_367,axiom,
! [A17: $tType] : ( lattice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_368,axiom,
! [A17: $tType] : ( order @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_369,axiom,
! [A17: $tType] : ( top @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_370,axiom,
! [A17: $tType] : ( ord @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_371,axiom,
! [A17: $tType] : ( bot @ ( filter @ A17 ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_372,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( finite_finite @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_373,axiom,
! [A17: $tType] : ( size @ ( option @ A17 ) ) ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_374,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_375,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_376,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_377,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_378,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_379,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_380,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_381,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_382,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_383,axiom,
real_V5047593784448816457lgebra @ 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_Ometric__space_392,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_393,axiom,
topolo1287966508704411220up_add @ 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___Limits_Otopological__group__add_401,axiom,
topolo1633459387980952147up_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_402,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_403,axiom,
topological_t1_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_404,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_405,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_406,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_407,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_408,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_409,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_410,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_411,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_412,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_413,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_414,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_415,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_416,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_417,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_418,axiom,
comm_semiring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_419,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_420,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_421,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_422,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_423,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_424,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_425,axiom,
idom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_426,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_427,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_428,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_429,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_430,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_431,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_432,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_433,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_434,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_435,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_436,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom_437,axiom,
semidom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_438,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_439,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_440,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_441,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_442,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_443,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_444,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_445,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_446,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_447,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_448,axiom,
dvd @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_449,axiom,
condit6923001295902523014norder @ 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___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_456,axiom,
linord4140545234300271783up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_457,axiom,
comple6319245703460814977attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_458,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_459,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_460,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_461,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_462,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_463,axiom,
comple9053668089753744459l_ccpo @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_464,axiom,
semilattice_sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_465,axiom,
semilattice_inf @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_466,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_467,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_468,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_469,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_470,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_471,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_472,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_473,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_OSup_474,axiom,
complete_Sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_OInf_475,axiom,
complete_Inf @ 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___Rings_Ozero__neq__one_484,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_485,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_486,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_487,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_488,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_489,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_490,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_491,axiom,
lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_492,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_493,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_494,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Otop_495,axiom,
top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_496,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Obot_497,axiom,
bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_498,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_499,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_500,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_501,axiom,
plus @ 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,
! [A17: $tType,A18: $tType] :
( ( ( topolo4958980785337419405_space @ A17 )
& ( topolo4958980785337419405_space @ A18 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A17 @ A18 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_505,axiom,
! [A17: $tType,A18: $tType] :
( ( ( topological_t2_space @ A17 )
& ( topological_t2_space @ A18 ) )
=> ( topological_t2_space @ ( product_prod @ A17 @ A18 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_506,axiom,
! [A17: $tType,A18: $tType] :
( ( ( topological_t1_space @ A17 )
& ( topological_t1_space @ A18 ) )
=> ( topological_t1_space @ ( product_prod @ A17 @ A18 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_507,axiom,
! [A17: $tType,A18: $tType] :
( ( ( finite_finite @ A17 )
& ( finite_finite @ A18 ) )
=> ( finite_finite @ ( product_prod @ A17 @ A18 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_508,axiom,
! [A17: $tType,A18: $tType] : ( size @ ( product_prod @ A17 @ A18 ) ) ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_509,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_510,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_511,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_512,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_513,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_514,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_515,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_516,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_517,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_518,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_519,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_520,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_521,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_522,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_523,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_524,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_525,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_526,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_527,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_528,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_529,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_530,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_531,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_532,axiom,
euclid5891614535332579305n_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_533,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_534,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_535,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_536,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_537,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_538,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_539,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_540,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_541,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_542,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_543,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_544,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_545,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_546,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_547,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_548,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_549,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_550,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_551,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_552,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_553,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_554,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_555,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_556,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_557,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_558,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_559,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_560,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_561,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_562,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_563,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_564,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_565,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_566,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_567,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_568,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_569,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_570,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_571,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_572,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_573,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_574,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_575,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_576,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_577,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_578,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_579,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_580,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_581,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_582,axiom,
idom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_583,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_584,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_585,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_586,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_587,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_588,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_589,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_590,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_591,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_592,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_593,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_594,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_595,axiom,
semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_596,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_597,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_598,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_599,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_600,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_601,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_602,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_603,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_604,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_605,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_606,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_607,axiom,
dvd @ code_integer ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_608,axiom,
size @ vEBT_VEBT ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X8: A] : ( P @ X8 ) ) ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
! [X6: nat] :
( ( member @ nat @ X6 @ ( set2 @ nat @ xs ) )
=> ( ord_less @ nat @ X6 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ n ) ) ) ).
thf(conj_1,hypothesis,
ord_less @ nat @ ( zero_zero @ nat ) @ n ).
thf(conj_2,conjecture,
( ( vEBT_Intf_test @ n @ xs @ ys )
= ( map @ nat @ nat
@ ^ [Y4: nat] :
( order_Greatest @ nat
@ ^ [Y7: nat] :
( ( member @ nat @ Y7 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( set2 @ nat @ xs ) ) )
& ( ord_less_eq @ nat @ Y7 @ Y4 ) ) )
@ ys ) ) ).
%------------------------------------------------------------------------------